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REVISION on Parabola & Ellipse

Q.1 Two mutually perpendicular tangents of the parabola y2 = 4ax meet the axis in P1 and P2. If S is the focus
1 1
of the parabola then  is equal to
l (SP1 ) l (SP2 )

4 2 1 1
(A) (B) (C) (D)
a a a 4a

Q.2 Which one of the following equations represented parametrically, represents equation to a parabolic
profile ?
t
(A) x = 3 cos t ; y = 4 sin t (B) x2  2 =  2 cos t ; y = 4 cos2
2
t t
(C) x = tan t ; y = sec t (D) x = 1  sin t ; y = sin + cos
2 2

x2 y2
Q.3 Let 'E' be the ellipse + = 1 & 'C' be the circle x2 + y2 = 9. Let P & Q be the points (1 , 2) and
9 4
(2, 1) respectively. Then :
(A) Q lies inside C but outside E (B) Q lies outside both C & E
(C) P lies inside both C & E (D) P lies inside C but outside E.

Q.4 Let S be the focus of y2 = 4x and a point P is moving on the curve such that it's abscissa is increasing at
the rate of 4 units/sec, then the rate of increase of projection of SP on x + y = 1 when P is at (4, 4) is
3
(A) 2 (B) – 1 (C) – 2 (D) –
2

Q.5 The points of contact Q and R of tangent from the point P (2, 3) on the parabola y2 = 4x are
1
(A) (9, 6) and (1, 2) (B) (1, 2) and (4, 4) (C) (4, 4) and (9, 6) (D) (9, 6) and ( , 1)
4
y2
Q.6 The eccentricity of the ellipse (x – 3)2 + (y – 4)2 = is
9
3 1 1 1
(A) (B) (C) (D)
2 3 3 2 3

Q.7 A tangent is drawn to the parabola y2 = 4x at the point 'P' whose abscissa lies in the interval [1,4]. The
maximum possible area of the triangle formed by the tangent at 'P' , ordinate of the point 'P' and the
x-axis is equal to
(A) 8 (B) 16 (C) 24 (D) 32
Q.8 From an external point P, pair of tangent lines are drawn to the parabola, y2 = 4x. If 1 & 2 are the

inclinations of these tangents with the axis of x such that, 1 + 2 = , then the locus of P is :
4
(A) x  y + 1 = 0 (B) x + y  1 = 0 (C) x  y  1 = 0 (D) x + y + 1 = 0
x 2 y2
Q.9 For an ellipse   1 with vertices A and A', tangent drawn at the point P in the first quadrant meets
9 4
the y-axis in Q and the chord A'P meets the y-axis in M. If 'O' is the origin then OQ2 – MQ2 equals to
(A) 9 (B) 13 (C) 4 (D) 5

Q.10 Length of the normal chord of the parabola, y2 = 4x, which makes an angle of with the axis of x is:
4
(A) 8 (B) 8 2 (C) 4 (D) 4 2

Q.11 The coordiantes of the ends of a focal chord of a parabola y2 = 4ax are (x1, y1) and (x2, y2) then
x1x2 + y1y2 has the value equal to
(A) 2a2 (B) – 3a2 (C) – a2 (D) 4a2
x2 y 2
Q.12 The line, lx + my + n = 0 will cut the ellipse + = 1 in points whose eccentric angles differ by
a 2 b2
/2 if :
(A) a2l2 + b2n2 = 2 m2 (B) a2m2 + b2l2 = 2 n2
(C) a2l2 + b2m2 = 2 n2 (D) a2n2 + b2m2 = 2 l2

Q.13 Locus of the feet of the perpendiculars drawn from either foci on a variable tangent to the hyperbola
16y2 – 9x2 = 1 is
(A) x2 + y2 = 9 (B) x2 + y2 = 1/9 (C) x2 + y2 =7/144 (D) x2 + y2 = 1/16

Q.19 If the normal to a parabola y2 = 4ax at P meets the curve again in Q and if PQ and the normal at Q makes
angles  and  respectively with the x-axis then tan (tan  + tan ) has the value equal to
1
(A) 0 (B) – 2 (C) – (D) – 1
2
Q.20 If the normal to the parabola y2 = 4ax at the point with parameter t1 , cuts the parabola again at the point
with parameter t2 , then
(A) 2 < t 22 < 8 (B) 2 < t 22 < 4 (C) t 22 > 4 (D) t 22 > 8

x2 y2
Q.22 The equation to the locus of the middle point of the portion of the tangent to the ellipse + =1
16 9
included between the co-ordinate axes is the curve :
(A) 9x2 + 16y2 = 4 x2y2 (B) 16x2 + 9y2 = 4 x2y2
(C) 3x2 + 4y2 = 4 x2y2 (D) 9x2 + 16y2 = x2y2

Q.14 A parabola y = ax2 + bx + c crosses the x  axis at ( , 0) ( , 0) both to the right of the origin. A circle
also passes through these two points. The length of a tangent from the origin to the circle is :
bc b c
(A) (B) ac2 (C) (D)
a a a
Q.15 Two parabolas have the same focus. If their directrices are the x axis & the y  axis respectively, then
the slope of their common chord is :
(A) ± 1 (B) 4/3 (C) 3/4 (D) none

Q.16 A circle has the same centre as an ellipse & passes through the foci F1 & F2 of the ellipse, such that the
two curves intersect in 4 points. Let 'P' be any one of their point of intersection. If the major axis of the
ellipse is 17 & the area of the triangle PF1F2 is 30, then the distance between the foci is :
(A) 11 (B) 12 (C) 13 (D) none

Q.17 The straight line joining any point P on the parabola y2 = 4ax to the vertex and perpendicular from the
focus to the tangent at P, intersect at R, then the equaiton of the locus of R is
(A) x2 + 2y2 – ax = 0 (B) 2x2 + y2 – 2ax = 0
2 2
(C) 2x + 2y – ay = 0 (D) 2x2 + y2 – 2ay = 0

Q.18 A normal chord of the parabola y2 = 4x subtending a right angle at the vertex makes an acute angle 
with the x-axis, then  equals to
(A) arc tan 2 (B) arc sec 3 (C) arc cot 2 (D) none

Q.19 If the eccentricity of the hyperbola x2  y2 sec2  = 5 is 3 times the eccentricity of the ellipse
x2 sec2  + y2 = 25, then a value of  is :
(A) /6 (B) /4 (C) /3 (D) /2

Q.20 Point 'O' is the centre of the ellipse with major axis AB & minor axis CD. Point F is one focus of the
ellipse. If OF = 6 & the diameter of the inscribed circle of triangle OCF is 2, then the product
(AB) (CD) is equal to
(A) 65 (B) 52 (C) 78 (D) none

Q.21 Locus of the feet of the perpendiculars drawn from vertex of the parabola y2 = 4ax upon all such chords
of the parabola which subtend a right angle at the vertex is
(A) x2 + y2 – 4ax = 0 (B) x2 + y2 – 2ax = 0
2 2
(C) x + y + 2ax = 0 (D) x2 + y2 + 4ax = 0

Q.22 For all real values of m, the straight line y = mx + 9 m 2  4 is a tangent to the curve :
(A) 9x2 + 4y2 = 36 (B) 4x2 + 9y2 = 36 (C) 9x2  4y2 = 36 (D) 4x2  9y2 = 36
Q.23 C is the centre of the circle with centre (0, 1) and radius unity. P is the parabola y = ax2. The set of values
of 'a' for which they meet at a point other than the origin, is
 1 1 1 1 
(A) a > 0 (B) a   0,  (C)  ,  (D)  ,  
 2 4 2 2 
4 x2 y2
Q.24 A tangent having slope of  to the ellipse + = 1 intersects the major & minor axes in points A
3 18 32
& B respectively. If C is the centre of the ellipse then the area of the triangle ABC is :
(A) 12 sq. units (B) 24 sq. units (C) 36 sq. units (D) 48 sq. units
Q.25 TP & TQ are tangents to the parabola, y2 = 4ax at P & Q. If the chord PQ passes through the fixed point
( a, b) then the locus of T is :
(A) ay = 2b (x  b) (B) bx = 2a (y  a)
(C) by = 2a (x  a) (D) ax = 2b (y  b)

Q.26 Through the vertex O of the parabola, y2 = 4ax two chords OP & OQ are drawn and the circles on OP
& OQ as diameters intersect in R. If 1, 2 &  are the angles made with the axis by the tangents at P &
Q on the parabola & by OR then the value of, cot 1 + cot 2 =
(A)  2 tan  (B)  2 tan () (C) 0 (D) 2 cot 

Q.27 If the chord through the point whose eccentric angles are  &  on the ellipse,
(x2/a2) + (y2/b2) = 1 passes through the focus, then the value of (1 + e) tan(/2) tan(/2) is
(A) e + 1 (B) e  1 (C) 1  e (D) 0

Q.28 The given circle x2 + y2 + 2px = 0, p  R touches the parabola y2 = 4x externally, then
(A) p < 0 (B) p > 0 (C) 0 < p < 1 (D) p < – 1

Q.29 The tangent at P to a parabola y2 = 4ax meets the directrix at U and the latus rectum at V then SUV
(where S is the focus) :
(A) must be a right triangle (B) must be an equilateral triangle
(C) must be an isosceles triangle (D) must be a right isosceles triangle.

Q.30 Two parabolas y2 = 4a(x - l1) and x2 = 4a (y – l2) always touch one another, the quantities l1 and l2 are
both variable. Locus of their point of contact has the equation
(A) xy = a2 (B) xy = 2a2 (C) xy = 4a2 (D) none

Q.32 If a normal to a parabola y2 = 4ax make an angle  with its axis, then it will cut the curve again at an angle
1  1 
(A) tan–1(2 tan) (B) tan1  tan  (C) cot–1  tan  (D) none
2  2 

x2 y2 x2 y2
Q.33 Which one of the following is the common tangent to the ellipses, 2 2  =1& 2   =1?
a b b2 a a 2  b2
(A) ay = bx + a 4  a 2 b 2  b 4 (B) by = ax  a 4  a 2 b 2  b 4
(C) ay = bx  a 4  a 2 b 2  b 4 (D) by = ax + a 4  a 2 b 2  b 4

Q.34 The vertex of a parabola is (2,2) and the co-ordinates of its two extrimities of the latus rectum are (–2,0)
and (6,0). The equation of the parabola is
(A) y2 – 4y + 8x – 12 = 0
(B) x2 + 4x – 8y – 12 = 0
(C) x2 – 4x + 8y – 12 = 0
(D) x2 – 8y – 4x + 20 = 0

Q.35 The length of the chord of the parabola y2 = x which is bisected at the point (2, 1) is
(A) 2 3 (B) 4 3
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