Created by T.

Madas

CONIC
SECTIONS

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Created by T. Madas

CIRCLE

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 Created by T. Madas

Question 1 (***+)
A circle is given parametrically by the equations

x = 4 + 3cos θ , y = 3 + 3sin θ , 0 ≤ θ < 2π .

a) Find a Cartesian equation for the circle.

b) Find the equations of the two tangents to the circle, which pass through the
origin O .

( x − 4 )2 + ( y − 3)2 = 9 , y = 0 and y = 24 x
7

Created by T. Madas
 Created by T. Madas

Question 2 (****+)
The points A , B and C have coordinates ( 6,6 ) , ( 0,8 ) and ( −2,2 ) , respectively.

a) Find an equation of the perpendicular bisector of AB .

The points A , B and C lie on the circumference of a circle whose centre is located at
the point D .

b) Determine the coordinates of D .

MP1-W , y = 3 x − 2 , D ( 2, 4 )

Created by T. Madas
 Created by T. Madas

Question 3 (****+)

C1 C2

O C3
x

The figure above shows three circles C1 , C2 and C3 .

The coordinates of the centres of all three circles are positive.
• The circle C1 has centre at ( 6,6 ) and touches both the x axis and the y axis.

• The circle C2 has the same size radius as C1 and touches the x axis.

• The circle C3 touches the x axis and both C1 and C2 .

Determine an equation of C3 .

2
2
(
C2V , ( x − 12 ) + y − 3
2 ) =9
4

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 Created by T. Madas

Question 4 (****+)
A circle C has equation

x2 + y2 + 2 x − 4 y + 1 = 0

The straight line L with equation y = mx is a tangent to C .

Find the possible values of m and hence determine the possible coordinates at which
L meets C .

MP1-Y , m = 0, m = 4 ,
3 ( −1,0 ) , ( 53 , 54 )

Created by T. Madas
 Created by T. Madas

Question 5 (****+)
A circle C has equation

x 2 + y 2 + 4 x − 10 y + 9 = 0 .

a) Find the coordinates of the centre of C and the size of its radius.

A tangent to the circle T , passes through the point with coordinates ( 0, −1) and has
gradient m , where m < 0 .

b) Show that m is a solution of the equation

2m2 − 3m − 2 = 0 .

The tangent T meets C at the point P .

c) Determine the coordinates of P .

( −2,5 ) , r = 20 , P ( −4,1)

Created by T. Madas
 Created by T. Madas

Question 6 (****+)
A circle has equation

x 2 + y 2 − 4 x − 2 y = 13 .

a) Find the coordinates of the centre of the circle and the size of its radius.

The points A and B lie on the circle such that the length of AB is 6 units.

b) Show that
ACB = 90° , where C is the centre of the circle.

A tangent to the circle has equation y = k − x , where k is a constant.

c) Show clearly that

2 x 2 + 2 (1 − k ) x + k 2 − 2k − 13 = 0 .

d) Determine the possible values of k .

( 2,1) , r = 18 , k = −3, 9

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 Created by T. Madas

Question 7 (****+)

y
B

C
A
O
x

The figure above shows a circle whose centre is located at the point C ( k , h ) , where k
and h are constants such that 2 < h < 5 .

The points A ( 3, 2 ) and B ( 7,8 ) lie on this circle.

The straight line l passes through C and the midpoint of AB .

Given that the radius of the circle is 26 , find an equation for l , the value of k and
the value of h .

MP1-S , 2 x + 3 y = 25 , k = 8 , h = 3

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 Created by T. Madas

Question 8 (****+)
The straight line passing through the points P (1,9 ) and Q ( 5,5 ) is a tangent to a circle
with centre at C ( 6,8 ) .

Determine, in exact surd form, the radius of the circle.

In this question you may not use …

… a standard formula which determines the shortest distance of a point from
iiii a straight line.
… any form of calculus.

r= 8

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 Created by T. Madas

Question 9 (****+)
The straight line with equation y = 2 x − 3 is a tangent to a circle with centre at the
point C ( 2, −3) .

Determine, in exact surd form, the radius of the circle.

In this question you may not use …

… a standard formula which determines the shortest distance of a point from
iiii a straight line.
… any form of calculus.

MP1-U , r = 4 5
5

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 Created by T. Madas

Question 10 (*****)
The figure below shows the circle with centre at C (5,10) and radius 5.

y= 3x
4
C ( 5,10 )
P

O x

The straight lines with equations, x = 0 and y = 3 x are tangents to the circle at the
4
points P and Q respectively.

Show that the area of the triangle PCQ is 10 square units.

SP-B , proof

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 Created by T. Madas

Question 11 (*****)
A circle passes through the points A ( x1, y1 ) and A ( x2 , y2 ) .

Given that AB is a diameter of the circle, show that the equation of the circle is

( x − x1 )( x − x2 ) + ( y − y1 )( y − y2 ) = 0 .

SP-T , proof

Question 12 (*****)
A circle passes through the points P (18,0 ) and Q ( 32,0 ) . A tangent to this circle
passes through the point S ( 0,199 ) and touches the circle at the point T .

Given that the y axis is a tangent to this circle, determine the coordinates of T

SP-T , ( 49,31)

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 Created by T. Madas

Question 13 (*****)
The circle C1 has equation

x2 + y2 − 4 x − 4 y + 6 = 0 .

The circle C2 has equation

x 2 + y 2 − 10 x − 10 y + k = 0 ,

where k is a constant.

Given that C1 and C2 have exactly two common tangents, determine the range of
possible values of k .

SP-Y , 18 < k < 42

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 Created by T. Madas

Question 14 (*****) non calculator

C D
x
O

The figure above shows the circle with equation

x 2 + y 2 − 4 x − 8 y = 205 ,

with centre at the point C and radius r .

The straight line AB is parallel to the y axis and has length 24 units.

The tangents to the circle at A and B meet at the point D .

Find the length of AD and hence deduce the area of the kite CADB .

SP-E , AD = 20 , area = 300

Created by T. Madas
 Created by T. Madas

Question 15 (*****)
y

C2
O x
P
C1

The figure above shows a circle C1 with equation

x 2 + y 2 − 18 x + ky + 90 = 0 ,

where k is a positive constant.

a) Determine, in terms of k , the coordinates of the centre of C1 and the size of its
radius.

Another circle C2 has equation

x 2 + y 2 − 2 x − 2 y = 34 .

b) Given that C1 and C2 are touching externally at the point P , find …

i. … the value of k .

ii. … the coordinates of P .

2
( ) (
C2T , 9, − 1 k , r = k − 9 , k = 10 , P 29 , − 13
2 4 5 5 )

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 Created by T. Madas

Question 16 (*****)
The curve C has equation

y = x2 − 4 x + 7 .

The points P ( −1,12 ) and Q ( 4,7 ) lie on C .

The point R also lies on C so that
PRQ = 90° .

Determine, as exact surds, the possible coordinates of R .

C2S , ( 5+221 , 17+2 21 ) or ( 5−221 ,17−2 21 )

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 Created by T. Madas

Question 17 (*****)
A circle C is centred at ( a, a ) and has radius a , where a is a positive constant.

The straight line L has equation

4x − 3y + 4 = 0 .

Given that L is tangent to C at the point P , determine …

a) … an equation of C .

b) … the coordinates of P .

You may not use a formula which determines the shortest distance of a point from a
straight line in this question.

SP-W , ( x − 1)2 + ( y − 1)2 = 1 , ( )

P 1,8
5 5

Created by T. Madas
 Created by T. Madas

Question 18 (*****)
A curve in the x-y plane has equation

x 2 + y 2 + 6 x cosθ − 18 y sin θ + 45 = 0 ,

where θ is a parameter such that 0 ≤ θ < 2π .

Given that curve represents a circle determine the range of possible values of θ .

SP-I , { 14 π < θ < 43 π } ∪ { 54 π < θ < 74 π }

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 Created by T. Madas

Question 19 (*****)
The circles C1 and C2 have respective equations

( x + 1)2 + ( y + 2 )2 = 94 and ( x − 5 )2 + ( y − 6 )2 = 36 .

The point P lies on C2 so that the distance of P from C1 is least.

Determine the exact coordinates of P .

( )
SP-P , P 7 , 6
5 5

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 Created by T. Madas

Question 20 (*****)
The straight line L and the circle C , have respective equations

L : y = λ ( x − a) + a λ 2 +1 and C : x 2 + y 2 = 2ax ,

where a is a positive constant and λ is a parameter.

Show that for all values of λ , L is a tangent to C .

SPX-L , proof

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 Created by T. Madas

Question 21 (*****)
The straight line with equation

y = t ( x − 2) ,

where t is a parameter,

crosses the circle with equation

x2 + y 2 = 1

at two distinct points A and B .

a) Show that the coordinates of the midpoint of AB are given by

 2t 2 2t 
M ,− 2
.
 1+ t 2
1 + t 
 

b) Hence show that the locus of M as t varies is a circle, stating its radius and the
coordinates of its centre.

( x − 1)2 + y 2 = 1

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 Created by T. Madas

Question 22 (*****)
Two parallel straight lines, L1 and L2 , have respective equations

y = 2x + 5 and y = 2x −1 .

L1 and L2 , are tangents to a circle centred at the point C .

A third line L3 is perpendicular to L1 and L2 , and meets the circle in two distinct
points, A and B .

Given that L3 passes through the point ( 9,0 ) , find, in exact simplified surd form, the
coordinates of C .

SP-Q , C  1 5 + 61 , 1 15 + 61 
( ) ( )
10 5 

Created by T. Madas
 Created by T. Madas

Question 23 (*****)
Two circles, C1 and C2 , have respective radii of 4 units and 1 unit and are touching
each other externally at the point A .

The coordinates axes are tangents to C1 , whose centre P lies in the first quadrant.

The x axis is a tangent to C2 , whose centre Q also lies in the first quadrant.

The straight line l1 , passes through P and Q , and meets the x axis at the point R .

The straight line l2 has negative gradient, passes through R and is a common tangent
to C1 and C2 .

Determine, in any order and in exact form where appropriate, the coordinates of A , the
length of PR and an equation of l2 .

( )
SPX-O , A 36 , 8 , PR = 20 , 24 x + 7 y = 224
5 5 3

Created by T. Madas
 Created by T. Madas

Question 24 (*****)
A family of circles is passing through the points with coordinates ( 2,1) and ( 4,5)

Show that the equation of every such circle has equation

x 2 + y 2 + 2 x ( 2k − 9 ) + 2ky = 6k − 41 ,

where k is a parameter.

SP-H , proof

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 Created by T. Madas

Question 25 (*****)
Three circles, C1 , C2 and C3 , have their centres at A , B and C , respectively, so that
AB = 5 , AC = 4 and BC = 3 .

The positive x and y axis are tangents to C1 .

The positive x axis is a tangent to C2 .

C1 and C2 touch each other externally at the point M .

Given further that C3 touches externally both C1 and C2 , find, in exact simplified
form, an equation of the straight line which passes through M and C .

SPX-E , 5 y − 10 6 x + 36 + 30 6 = 0

Created by T. Madas
 Created by T. Madas

Question 26 (*****)
Two circles, C1 and C2 , are touching each other externally, and have respective radii
of 9 and 4 units.

A third circle C3 , of radius r , touches C1 and C2 externally.

Given further that all three circles have a common tangent, determine the value of r .

36
SPX-N , r = = 1.44
25

Created by T. Madas
 Created by T. Madas

Question 27 (*****)
The point A ( 6, −1) lies on the circle with equation

x2 + y2 − 4 x + 6 y = 7 .

The tangent to the circle at A passes through the point P , so that the distance of P
from the centre of the circle is 65 .

Another tangent to the circle, at some point B , also passes through P .

Determine in any order the two sets of the possible coordinates of P and B .

(
MP1-T , P ( 3,5 ) ∩ B − 1 , − 18
13 13 ) (
∪ P ( 9, −7 ) ∩ B 30 , − 97
13 13 )

Created by T. Madas
 Created by T. Madas

PARABOLA

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 Created by T. Madas

Question 1 (**)

( )
The general point P 9t 2 ,18t , where t is a parameter, lies on the parabola with
Cartesian equation

y 2 = 36 x .

a) Show that the equation of a tangent at the point P is given by

x − ty + 9t 2 = 0 .

The tangent to the parabola y 2 = 36 x at the point Q (1,6 ) crosses the directrix of the
parabola at the point D .

b) Find the coordinates of D .

SYNF-A , D ( −9, −24 )

Created by T. Madas
 Created by T. Madas

Question 2 (**)

y
y 2 = 28 x
P ( 7,14 )

O
x
F Q

not drawn to scale

The figure above shows the graph of the parabola with equation

y 2 = 28 x , x ∈  , x ≥ 0 .

The point P ( 7,14 ) lies on the parabola.

a) Find an equation of the normal to the parabola at P .

This normal meets the x axis at the point Q and F is the focus of the parabola.

b) Determine the area of the triangle PQF .

x + y = 21 , area = 98

Created by T. Madas
 Created by T. Madas

Question 3 (**+)
A parabola H has Cartesian equation

y 2 = 12 x , x ≥ 0 .

( )
The point P 3t 2 ,6t , where t is a parameter, lies on H .

a) Show that the equation of a tangent to the parabola at P is given by

yt = x + 3t 2 .

The tangent to the parabola at P meets the y axis at the point Q and the point S is
the focus of the parabola.

b) Show further that …

i. … PQ is perpendicular to SQ .

(
ii. … the area of the triangle PQS is 9 t 1 + t 2 .
2 )
FP3-M , proof

Created by T. Madas
 Created by T. Madas

Question 4 (***)

( )
The general point P 3t 2 ,6t lies on a parabola.

a) Show that the equation of a tangent at P is given by

ty = x + 3t 2 .

The point Q ( −12,9 ) does not lie on the parabola.

b) Find the equations of the two tangents to the parabola which pass through Q
and deduce the coordinates of their corresponding points of tangency.

x + y + 3 = 0, ( 3, −6 ) , 4 y = x + 48, ( 48, 24 )

Created by T. Madas
 Created by T. Madas

Question 5 (***)

( )
The general point P 2t 2 , 4t lies on a parabola.

a) Show that the equation of a normal at P is given by

y + tx = 4t + 2t 3 .

b) Find the equation of each of the three normals to the parabola that meet at the
point with coordinates (12,0 ) .

y = 0, y + 2 x = 24, y − 2 x = −24

Created by T. Madas
 Created by T. Madas

Question 6 (***)
A parabola is defined parametrically by

x = at 2 , y = 2at , t ∈  ,

where a is a positive constant and t is a parameter.

a) Show that an equation of a normal to the parabola at the point P , where t = p ,

p ≠ 0 , is given by

y + px = 2ap + ap3 .

The normal at P meets the x axis at the point Q .

The point R , lies on the x axis, so that PR is parallel to the y axis.

b) Show that the distance QR remains constant for all values of the parameter,
and state this distance.

QR = 2a

Created by T. Madas
 Created by T. Madas

Question 7 (***+)

( )
The point P 4 p 2 ,8 p , p ≥ 0 , lies on the parabola with equation

y 2 = 16 x , x ≥ 0 .

a) Show that the equation of the tangent to the parabola at P is given by

yp = x + 4 p 2 .

The tangent to the parabola at P meets the directrix of the parabola at the point A and
the x axis at the point B . The point F is the focus of the parabola.

b) Given that the y coordinate of A is 42 , find the area of the triangle FBP .
5

FP3-J , area = 290

Created by T. Madas
 Created by T. Madas

Question 8 (***+)

( )
The point P 3 p 2 ,6 p , p > 0 , lies on the parabola with equation

y 2 = 12 x , x ≥ 0 .

a) Show that the equation of the normal to the parabola at P is given by

y + px = 6 p + 3 p3 .

The normal to the parabola at P meets the x axis at the point A and the directrix of
the parabola at the point B . The point C is the point of intersection of the directrix of
the parabola with the x axis.

b) Given that the coordinates of A are ( 223 ,0) , find as an exact simplified
fraction the area of the triangle BCP .

403
area =
27

Created by T. Madas
 Created by T. Madas

Question 9 (***+)
A parabola has its focus at T (1,1) and its directrix has equation x − 7 = 0 .

a) Find an equation for the parabola.

b) Sketch the parabola and show that its latus rectum is 12 units.

( y − 1)2 = 12 ( 4 − x )

Created by T. Madas
 Created by T. Madas

Question 10 (***+)
A parabola is given parametrically by the equations

x = 4 − t2 , y = 1− t , t ∈  .

a) Show that an equation of the normal at the general point on the parabola is

y + 2tx = 1 + 7t − 2t 3 .

The normal to parabola at P ( 3,0 ) meets the parabola again at the point Q .

b) Find the coordinates of Q .

( )
Q 7,5
4 2

Created by T. Madas
 Created by T. Madas

Question 11 (***+)
The points P and Q lie on the parabola with equation

y2 = 2x ,

so that OP is perpendicular to OQ , where O is the origin.

The point M is the midpoint of PQ .

Show that the Cartesian locus of M lies on the curve with equation

y2 = x − 2 .

FP3-I , proof

Created by T. Madas
 Created by T. Madas

Question 12 (***+)
The point P has coordinates

( )
P at 2 , 2at ,

where a is a positive constant and t is a real parameter.

The point P traces a parabola.

a) Show that the equation of a normal at the point P is given by

y + tx = 2at + at 3 .

b) Show that the straight line with equation

y = 2 x − 12a ,

is the only normal to the parabola passing through the point Q ( 3a, −6a ) .

c) Determine the coordinates of the two points of intersection between this normal
and the parabola, indicating clearly which point of intersection represents the
point of normality.

( 9 a, 6a ) , normal at ( 4a, −4a )

Created by T. Madas
 Created by T. Madas

Question 13 (***+)
A straight line L is a tangent to the parabola with equation

y 2 = Ax

where A is a positive constant.

Given that L does not pass through the origin O , show that the product of the gradient
and the y intercept of L equals the x coordinate of the focus of the parabola.

proof

Created by T. Madas
 Created by T. Madas

Question 14 (***+)
Sketch the parabola with equation

y2 − 4 y − 2x = 2 .

The sketch must include the …

a) … coordinates of points of intersection with the coordinate axes.

b) … coordinates of the vertex of the parabola.

c) … coordinates of the focus of the parabola.

d) … equation of the directrix of the parabola.

FP3-W , graph

Created by T. Madas
 Created by T. Madas

Question 15 (***+)

y
y 2 = 4ax
P
Q

O x
F

The figure above shows the sketch of the parabola with equation

y 2 = 4ax ,

where a is a positive constant.

The straight line L and the point F are the directrix and the focus of the parabola,
respectively.

The point P ( 8, y ) , y > 0 , lies on the parabola. The point Q lies on L , so that QP is
parallel to the x axis.

Given further that PF = 10 , determine the area of the triangle FPQ .

area = 40

Created by T. Madas
 Created by T. Madas

Question 16 (***+)

( )
The point T at 2 , 2at , lies on the parabola with equation

y 2 = 4ax, a > 0 , x ≥ 0 .

a) Show clearly that an equation of a normal to the parabola at the point

( )
P ap 2 , 2ap , p ≠ 0 , can be written as

y + px = 2ap + ap3 .

The normal at P meets the x axis at the point Q .

The midpoint of PQ is M .

b) Show that the locus of M as p varies is the parabola with equation

y2 = a ( x − a ) .

c) Find the coordinates of the focus of y 2 = a ( x − a ) .

( 54 a, 0)

Created by T. Madas
 Created by T. Madas

Question 17 (***+)

( )
The point T at 2 , 2at , where a is a positive constant and t is a real parameter, lies
on the parabola with equation

y 2 = 4ax .

A straight line passing through the origin, intersects at right angles the tangent to the
parabola at T , at the point P .

Show that as t varies, the Cartesian locus of P is

x3 + xy 2 + ay 2 = 0 .

FP3-G , proof

Created by T. Madas
 Created by T. Madas

Question 18 (****)
A parabola has its focus at the point with coordinates ( 4,1) and its directrix has
equation x = −1 .

Determine a Cartesian equation of the parabola.

( y − 1)2 = 10 ( x − 32 )

Question 19 (****)
A straight line L passes through the focus of the parabola with equation

y 2 = 12 x .

Given further that the shortest distance of L from the origin O is 12 , determine an
5
equation for L .

3 y = 4 x − 12

Created by T. Madas
 Created by T. Madas

Question 20 (****)
A parabola P has Cartesian equation

y 2 − 4 y − 8 x + 28 = 0 .

a) Determine …

i. … the coordinates of the vertex of P .

ii. … the coordinates of the focus of P .
iii. … the equation of the directrix of P .

The line with equation

y = mx + 1 , where m is a constant,

is a tangent at some point of P .

b) Find the possible values of m .

vertex at ( 3,2 ) , focus at ( 5,2 ) , directrix x = 1 , m = − 2 , 1

3

Created by T. Madas
 Created by T. Madas

Question 21 (****)

C y = 1 x2
y 12

L A

P
O
x

1 2
The figure above shows the parabola C with equation y = x .
12

The dotted line in the figure is the reflection of C in the line y = x .

a) Find the exact distance between the focus of C and the focus of its reflection.

The parabola intersects its reflection at the origin and at the point A .

b) Determine the coordinates of A .

The straight line L is a common tangent to both C and the reflection of C .

L touches C at the point P and the reflection of C at the point Q .

c) Determine the coordinates of P and Q .

FP3-P , 3 2 , A (12,12 ) , P ( −6,3) , Q ( 3, −6 )

Created by T. Madas
 Created by T. Madas

Question 22 (****)

( )
The point T at 2 , 2at , lies on the parabola with equation

y 2 = 4ax, a > 0 , x ≥ 0 .

a) Show clearly that an equation of a normal to the parabola at the point

( )
P ap 2 , 2ap , p ≠ 0 , can be written as

y + px = 2ap + ap3 .

( )
The normal at P re-intersects the parabola at the point Q aq 2 , 2aq .

b) Show that

p2 + 2
q=− .
p

c) Given that the midpoint of PQ has coordinates ( 5a, −2a ) , find the value of p .

p =1

Created by T. Madas
 Created by T. Madas

Question 23 (****)

( )
The point P 2t 2 , 4t , lies on the parabola with equation

y 2 = 8x , x ≥ 0 .

a) Show that an equation of a tangent to the parabola at P , can be written as

yt = x + 2t 2 , t ≠ 0 .

The tangent to the parabola at P meets the y axis at the point A . The perpendicular
bisectors of the straight line segments AP and OA , meets at the point B .

b) Find the coordinates of B , in terms of t .

c) Sketch the locus of B as t varies.

(
B t 2 + 2, t )

Created by T. Madas
 Created by T. Madas

Question 24 (****)
A parabola C has Cartesian equation

y 2 + 4 y − 16 x + 36 = 0 .

a) Describe the transformations that map the graph of the curve with equation
y 2 = 16 x onto the graph of C .

b) Determine the coordinates of the focus of C .

c) Show that …

( )
i. … the point P 4t 2 + 2,8t − 2 , lies on the parabola.

ii. … the equation of a tangent to the parabola at the point P , is

yt = x + 4t 2 − 2t − 2 .

d) Hence show that the gradients of the two tangents from the origin to the
parabola have gradients −2 and 1.

 2
translation by vector   , ( 6, −2 )
 −2 

Created by T. Madas
 Created by T. Madas

Question 25 (****)
Sketch the graph of the parabola with equation

y ( y − 2) = 4x + 3 .

The sketch must include the coordinates of any intersections with the axes and the
coordinates of the vertex of the parabola.

graph

Created by T. Madas
 Created by T. Madas

Question 26 (****)

( )
The point P ap 2 ,2ap , where p is a parameter, lies on the parabola, with Cartesian
equation

y 2 = 4ax ,

where a is a positive constant.

The point F is the focus of the parabola and O represents the origin.

The straight line which passes through P and F meets the directrix of the parabola at
15 2
the point Q , so that the area of the triangle OPQ is a .
4

Show that one of the possible values of p is 3 and find in exact surd form the other
2 possible values.

(
FP3-M , p = 1 3 ± 89
8 )

Created by T. Madas
 Created by T. Madas

Question 27 (****+)
A parabola P has focus S ( 6,0 ) and directrix the line x = 0 .

a) Show that a Cartesian equation for P is y 2 = 12 ( x − 3) .

b) Verify that the parametric equations of P are

x = 3t 2 + 3 , y = 6t .

(
c) Show that the equation of the tangent at the point Q 3q 2 + 3,6q is )
qy + 3 = x + 3q 2

The diagram below shows the parabola and its tangents at the points Q and R . The
point R lies on the parabola so that QSR is a straight line.

y
Q
y 2 = 12 ( x − 3)

x
O S
R

d) Show that the tangents to the parabola at Q and at R , meet on the y axis.

proof

Created by T. Madas
 Created by T. Madas

Question 28 (****+)
A parabola C has parametric equations

x = −2t 2 , y = 4t

a) Determine the coordinates of the focus and the equation of directrix of C .

( )
b) Show that an equation of the tangent to C , at the general point T −2t 2 , 4t is

yt + x = 2t 2

c) By considering the product of the roots of a suitable quadratic equation, show

that any two tangents that meet on the directrix of C , are perpendicular.

F ( −2, 0 ) , x = 2

Created by T. Madas
 Created by T. Madas

Question 29 (****+)

( )
The point P at 2 , 2at lies on the parabola with equation

y 2 = 4ax ,

where a is a positive constant and t is a real parameter.

( )
The normal to the parabola at P , meets the parabola again at the point Q as 2 , 2as .

Show that

16a 2 3
PQ =
t4
(t 2 + 1) .
proof

Created by T. Madas
 Created by T. Madas

Question 30 (****+)
A parabola C has Cartesian equation

y2 = 4x , x ∈  , x ≥ 0 .

( ) ( )
The points P p 2 , 2 p and Q q 2 , 2q are distinct and lie on C .

(
The tangent to C at P and the tangent to C at Q meet at R −1, 15 .
4 )
Calculate as an exact simplified fraction the area of the triangle PQR .

area = 4913
128

Created by T. Madas
 Created by T. Madas

Question 31 (****+)
A parabola C has Cartesian equation

y 2 = 4ax ,

where a is a positive constant.

( ) ( )
The points P ap 2 , 2ap and Q aq 2 , 2aq are distinct and lie on C .

The tangent to C at P and the tangent to C at Q meet at the point R .

Show that

2
SR = SP SQ ,

where S is the focus of the parabola.

SPX-F , proof

Created by T. Madas
 Created by T. Madas

Question 32 (****+)
A parabola has Cartesian equation

y = x2 , x ∈  .

A chord of the parabola is defined as the straight line segment joining any two distinct
points on the parabola.

Find the equation of the locus of the midpoints of parallel chords of the parabola whose
gradient is m .

SPX-D , x = 1 m
2

Created by T. Madas
 Created by T. Madas

Question 33 (*****)
The points P and Q have respective coordinates ( −1,6 ) and ( −5, −1) .

When the parabola with equation y = 4ax , where a is a constant, is translated by the
 −3 
vector   it passes through the point P .
 2

Find the possible values of the gradient of the straight line which passes through Q
and is a tangent to the translated parabola.

SPX-N , m = − 1 ∪ m = 2
2

Created by T. Madas
 Created by T. Madas

Question 34 (*****)
A parabola has Cartesian equation

y 2 = 12 x , x ≥ 0 .

The point P lies on the parabola and the point Q lies on the directrix of the parabola
so that PQ is parallel to the x axis.

The area of the triangle PQF is 8 2 square units, where the point F represents the
3
focus of the parabola.

Determine the coordinates of P , given further that the y coordinate of P is a positive

integer.

( )
SPX-I , P 4 , 4
3

Created by T. Madas
 Created by T. Madas

Question 35 (*****)

( )
The point P 2 p, p 2 , where p is a parameter, lies on the parabola, with Cartesian
equation

x2 = 4 y .

The point F is the focus of the parabola and O represents the origin.

The tangent to the parabola at P forms an angle θ with the positive x axis.

The straight line which passes through P and F forms an acute angle ϕ with the
tangent to the parabola at P .

Show that θ + ϕ = 1 π and hence state the coordinates of P if θ = ϕ .

2

SPX-I , P ( 2,1)

Created by T. Madas
 Created by T. Madas

Question 36 (*****)
A parabola has Cartesian equation

y = 1 x2 , x ∈  .
2

The points P and Q both lie on the parabola so that POQ is a right angle, where O is
the origin.

The point M represents the midpoint of PQ .

Show that as the position of P varies along the parabola, the locus of M is the curve
with equation

y = x2 − 2 .

FP3-T , proof

Created by T. Madas
 Created by T. Madas

Question 37 (*****)
The cubic equation

x3 + px + q = 0 ,

has 2 distinct real roots.

a) Show that 27q 2 + 4 p3 < 0 .

A parabola has Cartesian equation

y = x2 , x ∈  .

Three distinct normals to this parabola pass through the point, which does not lie on
the parabola, whose coordinates are ( a, b ) .

b) Show further that

2 ( )
b > 1 +3 1 a
4
3
.

SPX-M , proof

Created by T. Madas
 Created by T. Madas

Question 38 (*****)
A parabola is given parametrically by

x = 1 t2 , y = 2t , t∈ .
3 3

The normal to the parabola at the point P meets the parabola again at the point Q .

Show that the minimum value of PQ is 12 .

FP3-S , proof

Created by T. Madas
 Created by T. Madas

ELLIPSE

Created by T. Madas
 Created by T. Madas

Question 1 (**)
An ellipse has Cartesian equation

x2 y 2
+ =1.
16 9

a) Find the coordinates of its foci.

b) Sketch the ellipse.

(± 7,0 )

Question 2 (**)
An ellipse has parametric equations

x = 4cosθ , y = 7 sin θ , 0 ≤ θ < 2π .

a) Find the coordinates of its foci.

b) Sketch the ellipse.

( ±3, 0 )

Created by T. Madas
 Created by T. Madas

Question 3 (**)
25
An ellipse has a focus at ( 4,0 ) and the associated directrix has equation x = .
4

Determine a Cartesian equation of the ellipse.

x2 y2
+ =1
25 9

Question 4 (**)

x2
+ y2 = 1 .
4

The ellipse with Cartesian equation above and a parabola with vertex at the origin
share the same focal point.

Find the possible Cartesian equation for the parabola.

SYNF-B , y 2 = ± 48 x

Created by T. Madas
 Created by T. Madas

Question 5 (**+)
An ellipse E is given parametrically by the equations

x = cos t , y = 2sin t , 0 ≤ t < 2π .

a) Show that an equation of the normal to E at the general point P ( cos t , 2sin t )
can be written as

2y x
− =3.
sin t cos t

The normal to E at P meets the x axis at the point Q . The midpoint of PQ is M .

b) Find the equation of the locus of M as t varies.

x2 + y 2 = 1

Created by T. Madas
 Created by T. Madas

Question 6 (***)

x2 y 2
+ =1.
9 4

The ellipse with the Cartesian equation given above, has foci S and S ′ .

a) Find the coordinates S and S ′ .

b) Sketch the ellipse.

c) Show that for every point P on this ellipse,

SP + S ′P = 6 .

(± 5, 0 )

Created by T. Madas
 Created by T. Madas

Question 7 (***)
An ellipse E has Cartesian equation

x2 y 2
+ =1.
289 64

a) Find the coordinates of the foci of E , and the equations of its directrices.

b) Sketch the ellipse.

The point P lies on E so that PS is vertical, where S is the focus of the ellipse with
positive x coordinate.

c) Show that the tangent to the ellipse at the point P meets one the directrices of
the ellipse on the x axis.

( ±15,0 ) , x = ± 289
15

Created by T. Madas
 Created by T. Madas

Question 8 (***)
The point P ( 5cos θ , 4sin θ ) lies on the an ellipse E with Cartesian equation

16 x 2 + 25 y 2 = 400 .

a) Find the coordinates of the foci of E .

b) Show that an equation of the normal to the ellipse at P is

4 y cosθ − 5 x sin θ + 9sin θ cosθ = 0 .

The normal to the ellipse intersects the coordinate axes at the points A and B , and the
point M is the midpoint of AB .

c) Show that the locus of M , as θ varies, is the ellipse with equation

100 x 2 + 64 y 2 = 81 .

( ±3, 0 )

Created by T. Madas
 Created by T. Madas

Question 9 (***)
The ellipse E has parametric equations

x = a cos θ , y = b sin θ , 0 ≤ θ < 2π

where a and b are positive constants.

a) Show that an equation of the tangent at a general point on E is

bx cosθ + ay sin θ = ab .

This tangent to E intersects the coordinate axes at the points A and B , and the point
M is the midpoint of AB .

b) Find a Cartesian locus of M , as θ varies.

a2 b2
+ =1
ax 2 4 y2

Created by T. Madas
 Created by T. Madas

Question 10 (***)
An ellipse has Cartesian equation

x2 y 2
+ =1.
16 4

The general point P ( 4cos θ , 2sin θ ) lies on the ellipse.

a) Show that the equation of the normal to the ellipse at P is

2 x sin θ − y cosθ = 6sin θ cosθ .

The normal to the ellipse at P meets the x axis at the point Q and O is the origin.

b) Show clearly that as θ varies, the maximum area of the triangle OPQ is 4 1 .
2

proof

Created by T. Madas
 Created by T. Madas

Question 11 (***+)
An ellipse with equation

x2 y 2
+ =1
16 4

is transformed by the enlargement matrix E into a circle of radius 3 , with centre at the
origin.

Determine the elements of E .

3 0
SYNF-D , E =  4 
0 3
 2

Created by T. Madas
 Created by T. Madas

Question 12 (***+)
An ellipse has Cartesian equation

2 x 2 + 3 y 2 − 4 x + 12 y + 8 = 0 .

Determine …

a) … the coordinates of the centre of the ellipse.

b) … the eccentricity of the ellipse.

c) … the coordinates of the foci of the ellipse.

d) … the equations of the directrices of the ellipse.

(1, −2 ) , e = 3 , ( 0, −2 ) , ( 2, −2 ) , x = −2, x = 4
3

Created by T. Madas
 Created by T. Madas

Question 13 (***+)
An ellipse has Cartesian equation

x2 y2
+ = 1,
a2 12

where a is a positive constant.

The straight line with equation x = 8 is a directrix for the ellipse.

Determine the possible set of coordinates for the foci of the ellipse.

( ±2,0 ) or ( ±6,0 )

Created by T. Madas
 Created by T. Madas

Question 14 (***+)
An ellipse has equation

x 2 − 8 x + 4 y 2 + 12 = 0 .

a) Determine the coordinates of the foci and the equations of the directrices of the
ellipse.

A straight line with positive gradient passes through the origin O and touches the
ellipse at the point A .

b) Find the coordinates of A .

( ) ( )
FP3-K , 4 − 3, 0 , 4 + 3,0 , x = 4 − 4 3, x = 4 + 4 3 , 3, 1 2
3 3 3 ( )

Created by T. Madas
 Created by T. Madas

Question 15 (***+)
A point P lies on the ellipse with Cartesian equation

x2 y 2
+ =1.
64 16

The point Q is the foot of the perpendicular from the point P to the straight line with
equation x = 10 .

a) Sketch in the same diagram the ellipse, the straight line with equation x = 12
and the straight line segment PQ .

The point M is the midpoint of PQ .

b) Determine a Cartesian equation for the locus of M as the position of P varies,

further describing this locus geometrically.

FP3-N , ( x − 5)2 + y 2 = 16

Created by T. Madas
 Created by T. Madas

Question 16 (***+)
An ellipse E has Cartesian equation

x2 y 2
+ =1.
16 4

a) Show that an equation of the tangent to E at the point A ( 4cos θ , 2sin θ ) is

given by

2 y sin θ + x cosθ = 4 .

The point B ( 4cos θ , 4sin θ ) lies on the circle with Cartesian equation

x 2 + y 2 = 16 .

The tangent to the circle at the point B meets the tangent to the ellipse at the point A
at the point P .

b) Determine the coordinates of P , in terms of θ .

c) Describe mathematically the locus of P as θ varies.

FP3-Q , P ( 4secθ , 0 ) , the x axis, so that x ∈ ( −∞, −4] ∪ [ 4, ∞ )

Created by T. Madas
 Created by T. Madas

Question 17 (****)
An ellipse has Cartesian equation

x2
+ y2 = 1 .
2

A straight line L has equation y = mx + c , where m and c are positive constants.

a) Show that the x coordinates of the points of intersection between L and the
ellipse satisfy the equation

( 2m2 + 1) x2 + 4mcx + 2 ( c2 − 1) = 0 .
b) Given that L is a tangent to the ellipse, show that c 2 = 2m 2 + 1 .

The line L meets the negative x axis and the positive y axis at the points X and Y
respectively. The point O is the origin.

c) Find the area of the triangle OXY , in terms of m

d) Show that as m varies, the minimum area of the triangle OXY is 2.

e) Find the x coordinate of the point of tangency between the line L and the
ellipse when the area of the triangle is minimum.

1
area = m + , x = −1
2m

Created by T. Madas
 Created by T. Madas

Question 18 (****)
The point P ( x, y ) lies on an ellipse with foci at A ( 2,0 ) and B ( 6,0 ) .

Given further that

AP + BP = 10 ,

determine a simplified Cartesian equation for the ellipse, giving the final answer in the
form

f ( x, y ) = 1 .

( x − 4 )2 + y 2 =1
25 21

Created by T. Madas
 Created by T. Madas

Question 19 (****)
An ellipse has Cartesian equation

x2 y 2
+ =1.
25 9

The general point P ( 5cos θ ,3sin θ ) lies on the ellipse.

a) Show that the equation of the normal to the ellipse at P is

3 y cosθ − 5 x sin θ + 16sin θ cosθ = 0 .

The normal to the ellipse at P meets the x axis at the point Q and R is one of the
foci of the ellipse.

b) Show clearly that

QR
=e,
PR

where e is the eccentricity of the ellipse.

proof

Created by T. Madas
 Created by T. Madas

Question 20 (****+)
An ellipse is given, in terms of a parameter θ , by the equations

x = 3 2 cos θ , y = 4sin θ , 0 ≤ θ < 2π .

a) Determine …

i. … the coordinates of the foci of the ellipse.

ii. … the equations of the directrices of the ellipse.

b) Show that an equation of the tangent at a general point on the ellipse is

y sin θ x cos θ
+ = 1.
4 3 2

A straight line passes through the origin and meets the general tangent whose equation
is given in part (b), at the point P .

c) Show that, as θ varies, P traces the curve with equation

2
( x2 + y2 ) ( )
= 2 9x2 + 8 y 2 .

( )
F ± 2, 0 , x = ±9 2

Created by T. Madas
 Created by T. Madas

Question 21 (****+)
The equation of an ellipse is given by

x2 y 2
+ =1.
4 3

a) Determine the coordinates of the foci of the ellipse, and the equation of each of
its two directrices.

b) Show that

SP + TP = 4 .

A chord of the ellipse is defined as the straight line segment joining any two distinct
points on the ellipse.

c) Find the equation of the locus of the midpoints of parallel chords of the ellipse
whose gradient is 2 .

( ±1,0 ) , x = ±4 , y = − 1 x
8

Created by T. Madas
 Created by T. Madas

Question 22 (*****)
An ellipse has a focus at ( 5, −3) and directrix with equation y = 2 x − 7 .

5
Given that the eccentricity of the ellipse is , find the coordinates of the points of
10
intersection of the ellipse with the straight line with equation y = −3 .

V , FP3-U , ( 234 , −3) , ( 92 , −3)

Created by T. Madas
 Created by T. Madas

Question 23 (*****)
The point P lies on the ellipse with parametric equations

x = 3cos θ y = 2sin θ 0 ≤θ ≤ 1π .
2

The point M is the midpoint of PY , where Y is the point where the normal to ellipse
at P meets the y axis.

If O represents the origin, determine the maximum value of the area of the triangle
OMP , as θ varies.

V , SPX-Q , Area max 15

16

Created by T. Madas
 Created by T. Madas

Question 24 (*****)
The straight line L with equation y = mx + c , where m and c are constants, passes
through the point ( 25, 25 ) .

Given further that L is a tangent to the ellipse with equation

x2 y 2
+ =1,
25 9

determine the possible equations of L .

y = 4 x + 5 , y = 77 x − 85
5 60 12

Created by T. Madas
 Created by T. Madas

Question 25 (*****)
The point P lies on an ellipse whose foci are on the x axis at the points S and T .

Given further that the triangle STP is right angled at T , show that

1 − tan 1 θ
e= 2 ,
1 + tan 1 θ
2

where e is the eccentricity of the ellipse, and θ is the angle PST .

SPX-A , proof

Created by T. Madas
 Created by T. Madas

Question 26 (*****)
The point P lies on the ellipse with polar equation

r ( 5 − 3cos θ ) = 8 , 0 ≤ θ < 2π .

The ellipse has foci at O ( 0,0 ) and at T ( 3, 0 ) .

Show that OP + PT is constant for all positions of P .

SPX-D , OP + PT = 5

Created by T. Madas
 Created by T. Madas

RECTANGULAR
HYPERBOLA

Created by T. Madas
 Created by T. Madas

Question 1 (**)
The rectangular hyperbola H has Cartesian equation

xy = 9 , x ≠ 0 , y ≠ 0 .

 3
The point P  3t ,  , t ≠ 0 , where t is a parameter, lies on H .
 t

a) Show that the equation of a normal to H at P is given by

yt − xt 3 = 3 − 3t 4 .

The normal to H at the point where t = −3 meets H again at the point Q .

b) Determine the coordinates of Q .

( )
Q 1 ,81
9

Created by T. Madas
 Created by T. Madas

Question 2 (**)
The tangents to the hyperbola with equation xy = 9 , at two distinct points A and B ,
1
have gradient − .
16

Determine in any order …

a) … the coordinates of A and B .

b) … the equation of each of the two tangents.

( ) ( )
A 12, 3 , B −12, − 3 , x + 16 y = 24, x + 16 y = −24
4 4

Created by T. Madas
 Created by T. Madas

Question 3 (**+)
 4
The general point P  4t ,  , t ≠ 0 , where t is a parameter, lies on a hyperbola H .
 t

a) Show that the equation of a tangent at the point P is given by

x + t 2 y = 8t .

b) Find the equation of each of the two tangents to H which pass through the
point Q ( −12, 7 ) , and further deduce the coordinates of their corresponding
points of tangency.

x + 4 y = 16, ( 8, 2 ) , 49 x + 36 y + 336 = 0, ( − 247 , − 143 )

Created by T. Madas
 Created by T. Madas

Question 4 (***)
 c
The general point P  ct ,  , c > 0, t > 0 , lies on a hyperbola H with Cartesian
 t
equation

xy = c 2 .

The tangent to H at P meets the coordinate axes at the points A and B .

Given the area of the triangle BOA is 72 square units, find the value of c .

c=6

Created by T. Madas
 Created by T. Madas

Question 5 (***)
 3
The general point P  3t ,  , t ≠ 0 , where t is a parameter, lies on a hyperbola H .
 t

a) Show that the equation of a tangent at the point P is given by

x + t 2 y = 6t .

The tangents to the hyperbola at points A and B intersect at the point Q ( −1,7 ) .

b) Determine in any order …

i. …the coordinates of A and B .

ii. … the equation of each of the two tangents.

( )
A ( 3,3 ) , B − 3 , −21 , x + y = 6, 49 x + y + 42 = 0
7

Created by T. Madas
 Created by T. Madas

Question 6 (***+)
 a
The point P  ap,  lies on the rectangular hyperbola H , with Cartesian equation
 p

xy = a 2 ,

where a is a positive constant and p is a parameter.

a) Show that the equation of a tangent at the point P is given by

x + p 2 y = 2ap .

 a
The point Q  aq,  also lies on H , where q is a parameter, so that q ≠ p .
 q

The tangent at P and the tangent at Q intersect at the point R .

b) Find simplified expressions for the coordinates of R .

The values of p and q are such so that p = 3q .

c) Find a Cartesian locus of R as p varies.

 2apq 2a  3 2
R ,  , xy = 4 a
 p+q p+q

Created by T. Madas
 Created by T. Madas

Question 7 (****)
 c
The general point P  cp,  , p ≠ 0 , where p is a parameter, lies on the rectangular
 p
hyperbola, with Cartesian equation

xy = c 2 ,

where c is a positive constant.

a) Show that an equation of the tangent to the hyperbola at P is given by

yp 2 + x = 2cp .

 c
Another point Q  cq,  , p ≠ ± q also lies on the hyperbola.
 q

The tangents to the hyperbola at P and Q meet at the point R .

b) Show that the coordinates of R are given by

 2cpq 2c 
 , .
 p+q p+q

c) Given that PQ is perpendicular to OR , show that

p 2q 2 = 1 .

proof

Created by T. Madas
 Created by T. Madas

Question 8 (****)
The distinct points A , B and C lie on the hyperbola with equation

xy = p 2 ,

where p is a positive constant.

Given that ABC is a right angle, show that the tangent to the hyperbola at B , is
perpendicular to AC .

proof

Created by T. Madas
 Created by T. Madas

Question 9 (****+)
 5
The general point P  5t ,  , t ≠ 0 , where t is a parameter, lies on the hyperbola, with
 t
Cartesian equation

xy = 25 .

a) Show that an equation of the normal to the hyperbola at the point P is

5
y = t 2 x + − 5t 3
t

The normal to the hyperbola at P meets the hyperbola again at the point Q .

b) Show that the coordinates of Q are given by

 5 3
 − 3 , −5t  .
 t 

c) Show that the Cartesian form of the locus of the midpoint of PQ , as t varies,
is given by

2
 y x
4 xy + 25  −  = 0 .
 x y

proof

Created by T. Madas
 Created by T. Madas

Question 10 (****+)
 c
The general point P  cp,  , p ≠ 0 , where p is a parameter, lies on the rectangular
 p
hyperbola, with Cartesian equation

xy = c 2 ,

where c is a positive constant.

The normal to the hyperbola at P meets the hyperbola again at the point Q .

Show that the coordinates of Q are

 c 3
 − 3 , −cp  .
 p 

FP3-X , proof

Created by T. Madas
 Created by T. Madas

Question 11 (****+)
p 1 
The general point P  ,  , p ≠ 0 , where p is a parameter, lies on the rectangular
 2 2p 
hyperbola, with Cartesian equation

4 xy = 1 .

The normal to the hyperbola at P meets the hyperbola again at the point Q .

Show that the Cartesian form of the locus of the midpoint of PQ , as p varies, is

2
( y 2 − x2 ) + 16 x3 y 3 = 0 .

FP3-V , proof

Created by T. Madas
 Created by T. Madas

Question 12 (****+)
 2  2
Two distinct points P  2 p,  and Q  2q,  , lie on the hyperbola with Cartesian
 p  q
equation xy = 4 .

The tangents to the hyperbola at the points P and Q , meet at the point R .

a) Show that the coordinates of the point R are given by

4 pq 4
x= , y= .
p+q p+q

b) Given that the point R traces the rectangular hyperbola xy = 3 , find the two
possible relationships between p and q , in the form p = f ( q )

p = 3q , p = 1 q
3

Created by T. Madas
 Created by T. Madas

Question 13 (****+)
 2
The general point P  2t ,  , t ≠ 0 , where t is a parameter, lies on the rectangular
 t
hyperbola, with Cartesian equation

xy = 4 .

a) Find an equation of the normal to the hyperbola at the point P .

The normal to the hyperbola at P meets the hyperbola again at the point Q .

The point M is the midpoint of PQ .

b) Find an equation of the locus of M , as t varies.

Give a simplified answer in the form f ( x, y ) = 0 .

2
FP3-O , ty − 2 = t 3 x − 2t 4 , ( y 2 − x2 ) + x3 y 3 = 0

Created by T. Madas
 Created by T. Madas

Question 14 (*****)
 1 1
The point P  p + , p −  , p ≠ 0 , lies on the rectangular hyperbola, with Cartesian
 p p
equation

x2 − y 2 = 4 .

The normal to the hyperbola at P meets the y axis at the point Q ( 0, − k ) , k > 0 .

The area of the triangle OPQ , where O is the origin, is 15 .

4

Determine the two possible sets of coordinates for P .

SPX-C , ( 52 , − 32 ) , ( 52 , − 32 )

Created by T. Madas
 Created by T. Madas

Question 15 (*****)
The points P and Q are two distinct points which lie on the curve with equation

1
y= , x∈ , x ≠ 0 .
x

P and Q are free to move on the curve so that the straight line segment PQ is a
normal to the curve at P .

The tangents to the curve at P and Q meet at the point R .

Show that R is moving on the curve with Cartesian equation

2
( y2 − x2 ) + 4 xy = 0 .

SP-K , proof

Created by T. Madas
 Created by T. Madas

Question 16 (*****)
The variable point P lies on the rectangular hyperbola, with Cartesian equation

xy = a 2 ,

where a is a positive constant.

The normal to the hyperbola at P meets the hyperbola again at the point Q .

The point M is the midpoint of PQ .

Determine, in the form f ( x, y ) = 0 , an equation of the locus of M , for all the possible
positions of P .

2
(
SPX-E , a 2 y 2 − x 2 ) + 4 x3 y 3 = 0

Created by T. Madas
 Created by T. Madas

HYPERBOLA

Created by T. Madas
 Created by T. Madas

Question 1 (**)
A hyperbola H has foci at the points with coordinates ( −10,0 ) and (10,0 ) , and its
Cartesian equation is given by

x2 y2
− =1,
a2 36

where a is a positive constant.

a) Find the value of a .

b) Deduce the equations of the directrices of H .

a = 8 , x = ± 32
5

Created by T. Madas
 Created by T. Madas

Question 2 (**)
A hyperbola H has foci at the points with coordinates ( ±13,0 ) and the equations of its
144
directrices are x = ± .
13

Determine a Cartesian equation for H .

x2 y2
− =1
144 25

Created by T. Madas
 Created by T. Madas

Question 3 (***)
A hyperbola is given parametrically by

3  1 5  1
x = t +  , y = t − , t ≠ 0.
2 t  2 t 

a) Show that the Cartesian equation of the hyperbola can be written as

x2 y 2
− = 1.
9 25

b) Find …

i. … the equations of its asymptotes.

ii. … the coordinates of its foci.

iii. … the equations of its directrices.

c) Sketch the hyperbola indicating any intersections with the coordinate axes, as
well as the information stated in part (b).

SYNF-C , y = ± 5 x
3 (± )
34,0 , x = ± 9
34

Created by T. Madas
 Created by T. Madas

Question 4 (***)
The point P ( a secθ , b tan θ ) lies on the hyperbola with equation

x2 y2
− =1,
a2 b2

where a and b are positive constants.

a) Show that an equation of the normal at P is given by

( )
by + ax sin θ = a 2 + b 2 tan θ .

The normal to the hyperbola meets the coordinate axes at the points A and B .

b) Show that, as θ varies, the Cartesian locus of the midpoint of AB is given by

2
( ) (
4 a 2 x 2 − b2 y 2 = a 2 + b 2 ).
FP3-L , proof

Created by T. Madas
 Created by T. Madas

Question 5 (***+)
A hyperbola has Cartesian equation

2x2 − 4x − y2 − 4 y = 4 .

Find the coordinates of its foci and the equations of its directrices.

(1 + ) ( )
3, −2 , 1 − 3, −2 , x = − 1 + 1, x = 1 + 1
3 3

Created by T. Madas
 Created by T. Madas

Question 6 (****)
The general point P ( a secθ , b tan θ ) lies on the hyperbola with Cartesian equation

x2 y2
− =1,
a2 b2

where a and b are positive constants.

a) Show that an equation of the normal at P is given by

( )
by + ax sin θ = a 2 + b 2 tan θ .

The normal to the hyperbola meets the x axis at the point X .

The eccentricity of the hyperbola is 3 and its foci are denoted by S and S ′ , where S
2
has a positive x coordinate.

b) Given that OX = 3 OS , find the possible values of θ for 0 ≤ θ ≤ 2π .

π 5π
FP3-R , θ = ,
3 3

Created by T. Madas
 Created by T. Madas

Question 7 (****)
The equation of a hyperbola H is given in terms of a parameter t by

x = sinh t , y = cosh t , t ∈  .

a) Sketch the graph of H , clearly marking the equation of each of its asymptotes.

The equation of the tangent to H at the point P ( sinh t ,cosh t ) , meets each of the
asymptotes at the points A and B .

b) Show that P is equidistant from A and B .

c) Show further that the area of the triangle OAB , where O is the origin, is
exactly 1 square unit.

graph/proof

Created by T. Madas
 Created by T. Madas

Question 8 (****+)
A hyperbola H and a line L have the following Cartesian equations

x2 y2
H: − =1
a2 b2

L : y = mx + c ,

where a , b , m and c are non zero constants.

a) Show that the x coordinates of the points of intersection between L and H

satisfy the equation

(a m2 2
− b 2 ) x 2 + ( 2a 2mc ) x + a 2 ( b 2 + c 2 ) = 0 .

b) Given the line is a tangent to the hyperbola show that

a 2m2 = b 2 + c 2 .

c) Find the equations of the two tangents to the hyperbola with Cartesian equation

x2 y2
− =1
25 16

that pass through the point (1, 4 ) , and for each tangent the coordinates of their
point of tangency.

(
y = x + 3, − 25 16
) 4 16
(25
3 , − 3 , y = − 3 x + 3 , 4 , −3 )

Created by T. Madas
 Created by T. Madas

Question 9 (****+)
A hyperbola H has Cartesian equation

x2 y2
− =1,
a2 b2

where a and b are positive constants.

The straight line T1 is the tangent to H at the point ( a cosh θ , b sinh θ ) .

T1 meets the x axis at the point P .

The straight line T2 is a tangent to the hyperbola at the point ( a,0 ) .

T1 and T2 meet each other at the point Q .

Given further that M is the midpoint of PQ , show that as θ varies, the locus of M
traces the curve with equation

( )
x 4 y 2 + b 2 = ab 2 .

SPX-H , proof

Created by T. Madas
 Created by T. Madas

Question 10 (*****)
A hyperbola and an ellipse have respective equations

x2 y2 x2 y2
− =1 and + = 1,
a2 b2 a2 b2

where a > b > 0 .

The tangent to the hyperbola, at a point whose both coordinates are positive, passes
through the focus of the ellipse with positive x coordinate.

Show that the gradient of the above described tangent is 1 .

V , SPX-R , proof

Created by T. Madas