Question

A parabola of latus rectum $l$, touches a fixed equal parabola, the axes of curves being parallel. The locus of the vertex of moving curve is a parabola of latus rectum $kl$, then $k$ is equal to.

• A. 1/2
• B. 1
• C. 2
• D. 4

Answer 1

Answer: (C)

2

Answer 2

Let the fixed parabola be $y^2 = lx$. Its latus rectum is $l$. Let the moving parabola have vertex $(\alpha, \beta)$ and be equal to the fixed one, meaning its latus rectum is also $l$. For the locus of the vertex to be a parabola, its equation must be $(y-\beta)^2 = -l(x-\alpha)$. The condition for touching implies the discriminant of the resulting quadratic in $y$ is zero, leading to the locus $\beta^2 = 2l\alpha$. The locus of the vertex $(\alpha, \beta)$ is therefore $y^2 = 2lx$. The latus rectum of this locus parabola is $2l$. Comparing this with the given form $kl$, we find $kl = 2l$, which implies $k=2$.