Parabola Questions and Answers Part-2

1. Equation of a common tangent to the curves $y^{2}=8x$   and xy = – 1 is
a) 3y = 9x + 2
b) y = 2x + 1
c) 2y = x + 8
d) y = x + 2

Explanation: Equation of a tangent at (at2 , 2at) to y2 = 8x is ty = x + at2 where 4a = 8 i.e. a = 2

2. The tangent at the point $P\left(x_{1},y_{1}\right)$   to the parabola $y^{2}=4ax$   meets the parabola $y^{2}=4a$   (x + b) at Q and R, the coordinates of the mid-point of QR are
a) $\left(x_{1}-a,y_{1}+b\right)$
b) $\left(x_{1},y_{1}\right)$
c) $\left(x_{1}+b,y_{1}+a\right)$
d) $\left(x_{1}-b,y_{1}-b\right)$

Explanation: Equation of the tangent at P (x1, y1) to the

3. Consider a parabola y2 = 4ax, the length of focal chord is l and the length of the perpendicular from vertex to the chord is p then
a) l . p is constant
b) $l p^{2}$ is constant
c) $l^{2}$ p is constant
d) none of these

Explanation: Let P(at2 , 2at) and Q(a/t2 , –2a/t) be a focal

4. Tangent are drawn to a parabola from a point T. If P, Q are the points of contact, then perpendicular distance from P, T and Q upon the tangent at the vertex of the parabola are in
a) A.P
b) G.P
c) H.P
d) none of these

Explanation: Let P(at12 , 2at), Q(at22 , 2at2), then the point of intersection of the tangents t1y = x + at12 and t2y = x + at22 is T(at1t2, a(t1 + t2))

5. Chords of the parabola $y^{2}+4y=\frac{4}{3}x-\frac{16}{3}$     which subtend right angle at the vertex pass through
a) (7/3, –2)
b) (1/3, 0)
c) (4/3, 0)
d) (0, 4/3)

Explanation: Equation of the parabola is

6. The locus of the vertices of the family of parabolas $y=\frac{a^{3}x^{2}}{3}+\frac{a^{2}x}{2}-2a$
is
a) $xy=\frac{105}{64}$
b) $xy=\frac{3}{4}$
c) $xy=\frac{35}{16}$
d) $xy=\frac{64}{105}$

Explanation: Equation of the parabola is

7. Consider the two curves $c_{1}:y^{2}=4x;c_{2}:x^{2} +y^{2} -6x+1$
then
a) $c_{1}$ and $c_{2}$ touch each other only at one point
b) $c_{1}$ and $c_{2}$ touch each other exactly at two points
c) $c_{1}$ and $c_{2}$ intersect (but do not touch) at exactly two point
d) $c_{1}$ and $c_{2}$ neither intersect nor touch each other

Explanation: Solving the two equations, we get

8. If AB is a focal chord of the parabola $y^{2}=4ax$   with focus F, then harmonic mean FA and FB is
a) a
b) 2a
c) 4a
d) 8a

Explanation: End points of the focal chord be

9. Length of the shortest normal chord of the parabola $y^{2}=4x$   is
a) 6
b) $6\sqrt{3}$
c) 1
d) $3\sqrt{3}$

10. Let $\left(x_{1},y_{1}\right)$   be a point outside the parabola $y^{2}=4ax$   . Length of the chord of contact of tangents drawn from point $\left(x_{1},y_{1}\right)$   to $y^{2}=4ax$   is
a) $\frac{1}{a}\sqrt{\left(y_1^2-4ax_{1}\right)\left(y_1^2+4a^{2}\right)}$
b) $\frac{1}{a}\sqrt{\left(y_1^2-4ax_{1}\right)\left(x_1^2+a^{2}\right)}$
c) $\frac{1}{a}\sqrt{\left(y_1^2-4ax_{1}\right)\left(x_1^2+4a^{2}\right)}$
d) $\frac{1}{a}\sqrt{\left(y_1^2-4ax_{1}\right)\left(y_1^2+a^{2}\right)}$