Conic Section (Parabola-Ellipse-Hyperbola)

ASSIGNMENT-I (PARABOLA)

Q.1 Show that the normals at the points (4a, 4a) & at the upper end of the latus ractum of the
parabola y2 = 4ax intersect on the same parabola.
Q.2 Prove that the locus of the middle point of portion of a normal to y2 = 4ax intercepted
between the curve & the axis is another parabola. Find the vertex & the latus rectum of the
second parabola.
Q.3 Find the equations of the tangents to the parabola y2 = 16x, which are parallel &
perpendicular respectively to the line 2x – y + 5 = 0. Find also the coordinates of their points
of contact.
Q.4 A circle is described whose centre is the vertex and whose diameter is three-quarters of the
latus rectum of a parabola y2 = 4ax. Prove that the common chord of the circle and parabola
bisects the distance between the vertex and the focus.
Q.5 Find the length of latus rectum of the parabola 169{(x –1)2 + (y –3)2} = (5x – 12 y + 17)2.
Q.6 Through the vertex O of a parabola y2 = 4x, chords OP & OQ are drawn at right angles to one
another. Show that for all positions of P, PQ cuts the axis of the parabola at a fixed point.
Also find the locus of the middle point of PQ.
Q.7 Let S is the focus of the parabola y2 = 4ax and X the foot of the directrix, PP' is a double
ordinate of the curve and PX meets the curve again in Q. Prove that P'Q passes through focus.
Q.8 Three normals to y² = 4x pass through the point (15, 12). Show that one of the normals is
given by y = x  3 & find the equations of the others.
Q.9 Show that the locus of a point that divides a chord of slope 2 of a parabola y2 = 4x internally
in the ratio 1 : 2 is a parabola. Find the vertex of the parabola.
Q.10 Through the vertex O of the parabola y2 = 4ax, a perpendicular is drawn to any tangent
meeting it at P & the parabola at Q. Show that OP · OQ = constant.
Q.11 'O' is the vertex of the parabola y2 = 4ax & L is the upper end of the latus rectum. If LH is
drawn perpendicular to OL meeting OX in H, prove that the length of the double ordinate
through H is 4a 5 .
Q.12 The normal at a point P to the parabola y2 = 4ax meets its axis at G. Q is another point on the
parabola such that QG is perpendicular to the axis of the parabola.
Prove that QG2  PG2 = constant.
Q.13 If the normal at P(18, 12) to the parabola y2 = 8x cuts it again at Q, show that 9PQ = 80 10 .
Q.14 Prove that, the normal to y2 = 12x at (3, 6) meets the parabola again in (27, 18) & circle on
this normal chord as diameter is x2 + y2  30x + 12y  27 = 0.
Q.15 The abscissae of any two points on the parabola y2 = 4ax are in the ratio  : 1. Prove that the
locus of the point of intersection of tangents at these points is y2 = (1/4 + -1/4)2ax.
Q.16 P & Q are the points of contact of the tangents drawn from the point T to the parabola
y2 = 4ax. If PQ be the normal to the parabola at P, prove that TP is bisected by the directrix.
Q.17 Show that all chords of a parabola which subtend a right angle at the vertex pass through a
fixed point on the axis of the curve.
Q.18 From the point (1, 2) tangent lines are drawn to the parabola y2 = 4x. Find the equation of
the chord of contact. Also find the area of the triangle formed by the chord of contact & the
tangents.
Q.19 Prove that the circle described on any focal chord of a parabola as diameter touches the
directrix.

FIITJEE Limited, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi – 110016, Ph: 26515949, 26569493, Fax: 26513942 1
Q.20 From a point A common tangents are drawn to the circle x2 + y2 = a2/2 & parabola y2 = 4ax.
Find the area of the quadrilateral formed by the common tangents, the chord of contact of the
circle and the chord of contact of the parabola.
Q.21 Prove that on the axis of any parabola y² = 4ax there is a certain point K which has the
1 1
property that, if a chord PQ of the parabola be drawn through it, then  is
2
(PK) (QK) 2
same for all positions of the chord. Find also the coordinates of the point K.
Q.22 Prove that the two parabolas y2 = 4ax & y2 = 4c (x  b) cannot have a common normal, other
b
than the axis, unless > 2.
(a  c)
Q.23 Find the condition on ‘a’ & ‘b’ so that the two tangents drawn to the parabola y2 = 4ax from a
point are normals to the parabola x2 = 4by.
Q.24 Prove that the locus of the middle points of all tangents drawn from points on the directrix to
the parabola y2 = 4ax is y2(2x + a) = a(3x + a)2.
Q.25 Show that the locus of a point, such that two of the three normals drawn from it to the
parabola y2 = 4ax are perpendicular is y2 = a(x  3a).

ANSWER KEY
ASSIGNMENT-I
Q.2 (a, 0) ; a Q.3 2x  y + 2 = 0, (1, 4); x + 2y + 16 = 0, (16, 16)
28
Q.5 Q.6 (4 , 0) ; y2 = 2a(x – 4a)
13
2 8
Q.8 y = 4x + 72, y = 3x  33 Q.9  , 
9 9
Q.18 x  y = 1; 8 2 sq. units Q.20 15a2/ 4 Q.21 (2a, 0) Q.23 a2 > 8b2

FIITJEE Limited, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi – 110016, Ph: 26515949, 26569493, Fax: 26513942 2
 ASSIGNMENT-II (PARABOLA)
Q.1 In the parabola y2 = 4ax, the tangent at the point P, whose abscissa is equal to the latus
ractum meets the axis in T & the normal at P cuts the parabola again in Q.
Prove that PT : PQ = 4 : 5.
Q.2 Two tangents to the parabola y2= 8x meet the tangent at its vertex in the points P & Q. If
PQ = 4 units, prove that the locus of the point of the intersection of the two tangents is y2 = 8
(x + 2).
Q.3 A variable chord t1 t2 of the parabola y2 = 4ax subtends a right angle at a fixed point t0 of the
curve. Show that it passes through a fixed point. Also find the coordinates of the fixed point.
Q.4 Two perpendicular straight lines through the focus of the parabola y² = 4ax meet its directrix
in T & T' respectively. Show that the tangents to the parabola parallel to the perpendicular
lines intersect in the mid point of T T '.
Q.5 Two straight lines one being a tangent to y2 = 4ax and the other to x2 = 4by are right angles.
Find the locus of their point of intersection.
Q.6 A variable chord PQ of the parabola y2 = 4x is drawn parallel to the line y = x. If the
parameters of the points P & Q on the parabola are p & q respectively, show that p + q = 2.
Also show that the locus of the point of intersection of the normals at P & Q is 2x  y = 12.
Q.7 Show that an infinite number of triangles can be inscribed in either of the parabolas y2 = 4ax
& x2 = 4by whose sides touch the other.
Q.8 If (x1, y1), (x2, y2) and (x3, y3) be three points on the parabola y2 = 4ax and the normals at
x  x2 x  x 3 x 3  x1
these points meet in a point then prove that 1  2  = 0.
y3 y1 y2
Q.9 Show that the normals at two suitable distinct real points on the parabola y2 = 4ax intersect at
a point on the parabola whose abscissa > 8a.
Q.10 The equation y = x2 + 2ax + a represents a parabola for all real values of a.
(a) Prove that each of these parabolas pass through a common point and determine the
coordinates of this point.
(b) The vertices of the parabolas lie on a curve. Prove that this curve is a parabola and find its
equation.
Q.11 The normals at P and Q on the parabola y2 = 4ax intersect at the point R (x1, y1) on the
parabola and the tangents at P and Q intersect at the point T. Show that,
1
l(TP) · l(TQ) = (x1 – 8a) y12  4a 2
2
Also show that, if R moves on the parabola, the mid point of PQ lie on the parabola
y2 = 2a(x + 2a).
Q.12 If Q(x1, y1) is an arbitrary point in the plane of a parabola y2 = 4ax, show that there are three
points on the parabola at which OQ subtends a right angle, where O is the origin. Show
further that the normal at these three points are concurrent at a point R, determine the
coordinates of R in terms of those of Q.
Q.13 PC is the normal at P to the parabola y2 = 4ax, C being on the axis. CP is produced outwards
to Q so that PQ = CP; show that the locus of Q is a parabola, & that the locus of the
intersection of the tangents at P & Q to the parabola on which they lie is
y2 (x + 4a) + 16 a3 = 0.
Q.14 Show that the locus of the middle points of a variable chord of the parabola y2 = 4ax such that
the focal distances of its extremities are in the ratio 2 : 1, is 9(y2 – 2ax)2 = 4a2(2x – a)(4x + a).
Q.15 A quadrilateral is inscribed in a parabola y2 = 4ax and three of its sides pass through fixed
points on the axis. Show that the fourth side also passes through fixed point on the axis of the
parabola.
Q.16 Prove that the parabola y2 = 16x & the circle x2 + y2  40x  16y  48 = 0 meet at the point
P(36, 24) & one other point Q. Prove that PQ is a diameter of the circle. Find Q.

FIITJEE Limited, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi – 110016, Ph: 26515949, 26569493, Fax: 26513942 3
Q.17 A variable tangent to the parabola y2 = 4ax meets the circle x2 + y2 = r2 at P & Q. Prove that
the locus of the mid point of PQ is x(x2 + y2) + ay2 = 0.
Q.18 Find the locus of the foot of the perpendicular from the origin to chord of the parabola y2 =
4ax subtending an angle of 450 at the vertex.
Q.19 Show that the locus of the centroids of equilateral triangles inscribed in the parabola y2 = 4ax
is the parabola 9y2  4ax + 32 a2 = 0.
Q.20 The normals at P, Q, R on the parabola y2 = 4ax meet in a point on the line y = k. Prove that
the sides of the triangle PQR touch the parabola x2  2ky = 0.
Q.21 A fixed parabola y2 = 4 ax touches a variable parabola. Find the equation to the locus of the
vertex of the variable parabola. Assume that the two parabolas are equal and the axis of the
variable parabola remains parallel to the x-axis.
Q.22 Show that the circle through three points the normals at which to the parabola y2 = 4ax are
concurrent at the point (h, k) is 2(x2 + y2)  2(h + 2a) x  ky = 0.
Q.23 Prove that the locus of the centre of the circle, which passes through the vertex of the
parabola y2 = 4ax & through its intersection with a normal chord is 2y2 = ax  a2.
Q.24 The sides of a triangle touch y2 = 4ax and two of its angular points lie on y2 = 4b(x + c).
Show that the locus of the third angular point is a2y2 = 4(2b  a)2.(ax + 4bc)
Q.25 Three normals are drawn to the parabola y2 = 4ax cos  from any point lying on the straight
line y = b sin  Prove that the locus of the orthocentre of the triangles formed by the
x 2 y2
corresponding tangents is the ellipse 2  2 = 1, the angle  being variable.
a b

ANSWER KEY
ASSIGNMENT-II
Q.3 [a(t²o + 4),  2ato] Q.5 (ax + by) (x2 + y2) + (bx  ay)2 = 0

 1 1
Q.10 (a)   ,  ; (b) y = – (x2 + x) Q.12 ((x1 – 2a), 2y1) Q.21 y2 = 8 ax
 2 4

Q.16 Q(4, 8) Q.18 (x2 + y2 – 4ax)2 = 16a(x3 + xy2 + ay2)

FIITJEE Limited, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi – 110016, Ph: 26515949, 26569493, Fax: 26513942 4
FIITJEE Limited, 29-A, Kalu Sarai, Sarvapriya Vihar, New Delhi – 110016, Ph: 26515949, 26569493, Fax: 26513942 5