A N A LY T I C

GEOM 2
DISCUSSIONS ANALYTIC
GEOM 2

CONICS

Four ways of knowing conics (conic sections)

1. By cutting plane
2. By discriminant
3. By eccentricity
4. By equation
DISCUSSIONS ANALYTIC
GEOM 2

By Cutting Plane

• Parallel to the base: Circle

• Parallel to the element: Parabola
• Parallel to the axis: Hyperbola,
• Any angle provided not parallel to base,
element, and axis: Ellipse
DISCUSSIONS ANALYTIC
GEOM 2

General Equation of a Conic

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
The invariant B2-4AC can be used to test the type of conics represented by the equations
above. Thus,
1.Ellipse if B2 – 4AC < 0
2.Parabola if B2 – 4AC = 0
3.Hyperbola if B2 – 4AC > 0
DISCUSSIONS ANALYTIC
GEOM 2

Eccentricity Conics Eccentricity

Measurement of how “uncircle” the conic Circle 0
is
Ellipse <1
𝑓𝑜𝑐𝑢𝑠 Parabola 1
𝑒=
𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
Hyperbola >1
Straight Line ∞
DISCUSSIONS ANALYTIC
GEOM 2

By Equation

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

A=C : CIRCLE
A≠C, same sign ELLIPSE
A≠C, opposite sign HYPERBOLA
Either A or C is present PARABOLA
DISCUSSIONS ANALYTIC
GEOM 2

Principal Axis

Line or segment that divides conics into two equal parts through focus

Circle ::: Diameter

Parabola ::: Axis of Symmetry
Hyperbola ::: Transverse Axis
Ellipse ::: Major Axis
PROBLEM NO. 1 ANALYTIC
GEOM 2

What is the conic section formed by the equation

16x2-24xy+9y2-144x+8y=244?

A. Hyperbola
B. Parabola
C. Ellipse
D. Circle
DISCUSSIONS ANALYTIC
GEOM 2

Circles

Circle is the locus of point such that it moves equidistant from the fixed point called the
center.
General Equation of a Circle
Ax2 + Cy2 + Dx + Ey + F = 0
x2 + y2 + Dx + Ey + F = 0
Standard Equation of a Circle
(x – h)2 + (y – k)2 = r2
Center : (h, k)
DISCUSSIONS ANALYTIC
GEOM 2

The two circles C1 and C2 are said to be orthogonal (they intersect at right angles) if and
only if

D1D2 + E1E2 = 2(F1 + F2)

DISCUSSIONS ANALYTIC
GEOM 2

Radical axis

Radical Axis
Let the equation of the two intersecting
circles be as follows
C1 : x2 + y2 + D1x + E1y + F1 = 0 Line of Centers

C2 : x2 + y2 + D2x + E2y + F2 = 0 Radical axis

The line through the points of

intersection of C1 and C2 is called the
Line of Centers
radical axis of the circles and its
equations is
(D1 – D2)x + (E1 – E2)y + (F1 – F2) = 0
PROBLEM NO. 2 ANALYTIC
GEOM 2

Find the equation of the radical axis of the following circles

C1 : x2 + y2 – 5x + 3y – 2 = 0
C2 : x2 + y2 + 4x – y – 7 = 0

A. 9x + 4y + 5 = 0
B. 9x – 4y + 5 = 0
C. 9x – 4y – 5 = 0
D. 9x + 4y – 5 = 0
PROBLEM NO. 3 ANALYTIC
GEOM 2

Find the equation of the circle which has the line joining (4, 7) and (2, -3) as diameter.

A. (x-3)2+(y-2)2=26
B. (x+3)2+(y-2)2=26
C. (x+3)2+(y+2)2=26
D. (x-3)2+(y+2)2=26
PROBLEM NO. 4 ANALYTIC
GEOM 2

Find the radius and the equation of the line tangent to a circle whose center is (5,3) and
at the point (10, 15)

A. 13, 5x-12y=230
B. 13, 5x+12y=230
C. 15, 5x+12y=-230
D. 15, 5x-12y=-230
PROBLEM NO. 5 ANALYTIC
GEOM 2

Determine the equation of the circle whose radius is 5, center on the line x = 2 and
tangent to the line 3x – 4y + 11 = 0

A. (x-2)2+(y-2)2=5
B. (x-2)2+(y+2)2=25
C. (x-2)2+(y+2)2=5
D. (x-2)2+(y-2)2=25
DISCUSSIONS ANALYTIC
GEOM 2

Parabola

locus of a point which moves so that it is always equidistant from fixed point called focus
and a fixed line called directrix.

General Equation of a Parabola

Ax2 + Dx + Ey + F = 0 ::: Cy2 + Dx + Ey + F = 0
DISCUSSIONS ANALYTIC
GEOM 2

Standard Equation of a Parabola

(x – h)2 = ± 4a(y – k) (y – k)2 = ± 4a(x – h)

DISCUSSIONS ANALYTIC
GEOM 2

1.Distance from vertex to focus : a = VF

2.Length of the Latus Rectum : LR = 4a
3.Eccentricity :e=1
D
directrix Parabola (opens to the right)
L

2a Axis of the parabola

a
V F
Latus Rectum
R

D’
PROBLEM NO. 6 ANALYTIC
GEOM 2

Find the directrix of the parabola from the equation y-1=0.25 (x+2)2

A. y=2
B. y=0
C. x+y=0
D. x-y=0
PROBLEM NO. 7 ANALYTIC
GEOM 2

The parabola y = 3x2 – 6x + 5 has its vertex at.

A. (1, 2)
B. (1, 5)
C. (3, 2)
D. (3, 5)
PROBLEM NO. 8 ANALYTIC
GEOM 2

A parabolic arch, 18 meters high has a beam, 64 meters long placed along the arch 8
meters from the top. A beam is to be placed on the bottom. How long should this beam
be?

A. 48
B. 24
C. 69
D. 96
DISCUSSIONS ANALYTIC
GEOM 2

Ellipse
locus of point which moves so that the sum of its distance from two fixed points is
constant and is equal to the length of the major axis (2a).

General Equation of a Ellipse

Ax2 + Cy2 + Dx + Ey + F = 0
DISCUSSIONS ANALYTIC
GEOM 2

Standard Equation of a Ellipse

( x − h) 2 ( y − k ) 2 ( x − h) 2 ( y − k ) 2
1. 2
+ 2
=1 Center : (h, k) 2. 2
+ 2
=1 Center : (h, k)
a b b a
Major Axis : Horizontal Major Axis : vertical
DISCUSSIONS ANALYTIC
GEOM 2

2𝑏2
LR =
directrix
a directrix 𝑎

e = c/a=a/d < 1
B1
L1
L2 A = π ab
b
V2

c
a > b ::: a2 = b2 + c2
V1 F1 F2

R1
B2 R2 𝑎−𝑏
c Ellipse Flatness ::: 𝑓=
𝑎
′ 𝑎−𝑏
d Second Flatness ::: 𝑓 =
𝑏
𝑐
Second Eccentricity ::: 𝑒′ =
𝑏
PROBLEM NO. 9 ANALYTIC
GEOM 2

𝑥2 𝑦2
Find the values of a, b, c and e for the ellipse + = 1?
25 16

3
A. 5,4,3,
5
5
B. 3,4,5,
3
4
C. 4,3,5,
5
1
D. 1,2,3,
3
PROBLEM NO. 10 ANALYTIC
GEOM 2

Find the equation of the ellipse whose foci are the point (±2, 0), and one of whose
vertices is the point (3,0).

𝑥2 𝑦2
A. + =1
9 5
𝑥 2 𝑦2
B. + =1
3 5
𝑥 2 𝑦2
C. + =1
9 25
𝑥 2 𝑦2
D. − =1
9 5
PROBLEM NO. 11 ANALYTIC
GEOM 2

An ellipse has its vertices at (-2, -3) and (8, -3). If one of the minor axis is at (3, -7), how
far is the nearest focus to the left of the directrix?

A. 4.33
B. 5.33
C. 6.33
D. 7.33
DISCUSSIONS ANALYTIC
GEOM 2

Hyperbola

locus of a point which moves so that the

difference of the distances from two fixed
points (foci) is constant and is equal to the
length of the transverse axis (2a).

EQUILATERAL/ RECTANGULAR HYPERBOLA

Transverse axis and conjugate axis have the
same length and asymptotes are
perpendicular to each other.

General Equation of a Hyperbola

±Ax2 + ∓ Cy2 + Dx + Ey + F = 0
DISCUSSIONS ANALYTIC
GEOM 2

Standard Equation of a Hyperbola

DISCUSSIONS ANALYTIC
GEOM 2

asymptotes
1.a = semi transverse axis CV1 = CV2
2.b = semi conjugate axis CB1 = CB2
B1 L2 3.c = distance from center to focus = CF1 = CF2
L1
4.2a = transverse axis = V1V2
b
5.2b = conjugate axis B1B2

R1 V1 V2 F2 6.2c = distance from focus to focus = F1F2

C
7.a < b, a = b, or a > b
R1
B2 R2
8.a2 + b2 = c2
d 9.eccentricity : e = c/a > 1
directrices
a 10.Distance from center to directrix : d = a/e
11.LR = 2b2/a = length of latus rectum
c
PROBLEM NO. 12 ANALYTIC
GEOM 2

What are the coordinates of the foci of the hyperbola

𝑥 2 -4𝑦 2 =4

A. (±5,0)
B. (±2,0)
C. (±3,0)
D. (±√5,0)
PROBLEM NO. 13 ANALYTIC
GEOM 2

Find the equation of the asymptotes of the hyperbola

𝑥 2 -4𝑦 2 =4

A. 2𝑦 ±x=0
B. 2𝑦 ±x=4
C. 2𝑦 ±x=1
D. 2𝑦 ±x=2
PROBLEM NO. 14 ANALYTIC
GEOM 2

The eccentricity of the hyperbola 6(y – 6)2 – 9(x – 7)2 = 144 is equal to

A. 6/7
B. 5/3
C. 9/7
D. 6/9
DISCUSSIONS ANALYTIC
GEOM 2

Translation of Axis

Wherein (x,y) is the original point and (h,k) is the new origin

x’ = x-h
y ’=y-k
DISCUSSIONS ANALYTIC
GEOM 2

Rotation of Axis

x’ = xcos 𝜃 + y sin 𝜃
y’=-xsin 𝜃 +ycos 𝜃
DISCUSSIONS ANALYTIC
GEOM 2

Reflection of Axis
PROBLEM NO. 15 ANALYTIC
GEOM 2

Find the coordinates of the points (2,3), (-5,7) and (0,2) in a system of coordinates
whose origin is the point (1,3).

A. (1,0), (-6,4), (-1,-1)

B. (1,0), (-6,4), (1,-1)
C. (1,1), (-6,4), (-1,-1)
D. (-1,0), (-6,4), (-1,-1)
PROBLEM NO. 16 ANALYTIC
GEOM 2

Find the coordinates the points (1,0), (2,2) and (-3,4) referred to a system of rectangular
coordinates obtained by turning the coordinate axes through an angle of 450.

2 − 2 2 7 2
A. , , 2 2, 0 , ( , )
2 2 2 2
2 − 2 2 7 2
B. , , 0,0 , ( , )
2 2 2 2
2 2 2 7 2
C. , , 2 2, 0 , ( , )
2 2 2 2
− 2 2 2 7 2
D. , , 2 2, 0 , ( , )
2 2 2 2
DISCUSSIONS ANALYTIC
GEOM 2

POLAR TO RECTANGULAR
Polar Coordinates 𝑥 = 𝑟𝑐𝑜𝑠𝜃 𝑦 = 𝑟𝑠𝑖𝑛𝜃
•θ is positive if measured counterclockwise RECTANGULAR TO POLAR
•θ is negative if measured clockwise
𝑟= 𝑥2 + 𝑦2
𝑦
𝑡𝑎𝑛𝜃 =
𝑥
Radius vector
DISCUSSIONS ANALYTIC
GEOM 2

The distance between two points (r1, θ1) and (r2, θ2) can be found using cosine law.

𝑑= 2 2
𝑟1 + 𝑟2 − 2𝑟1 𝑟2 𝑐𝑜𝑠𝜃
PROBLEM NO. 17 ANALYTIC
GEOM 2

Calculate the distance between (3,60∘) and (5,145∘).

A. 6.90
B. 5.60
C. 4.70
D. 3.90
PROBLEM NO. 18 ANALYTIC
GEOM 2

Convert r(-2 sin𝜃+ 3 cos𝜃) = 2 into rectangular form.

A. -2y + 3x = 2
B. -2y - 3x = 2
C. -3y + 2x = 2
D. 3y - 2x = 2
PROBLEM NO. 19 ANALYTIC
GEOM 2

Convert 𝜃 + 𝜋/4 = 0 into rectangular form.

A. y=x
B. y=-x
C. y+1=x
D. y-1=x
PROBLEM NO. 20 ANALYTIC
GEOM 2

Given the polar equation r = sin 𝜃. Determine the rectangular coordinates (x,y) of a
point in the curve when 𝜃 is 30 degrees

A. (2.17, 1.25)
B. (3.08, 1.5)
C. (2.51, 4.12)
D. (6, 3)