Question

The locus of the mid point of the focal radii of a varable point moving on the parabola , $y^2=4ax$ is a parabola whose

• A. Latus rectum is half the latus rectum of the original parabola.
• B. Vertex is $(a/2,0)$
• C. Directrix is y-axis.
• D. Focus has the coordinates $(a,0)$

Answer 1

Answer: (B)

Vertex is $(a/2,0)$

Equation of parabola $=y^2=4ax$

Any point on the parabola is $(a{t}^2,2at)$ is

$(\frac{a+a{t}^2}{2},at)$

$\text{Then,}$

$\text{n}\text{=}\frac{a+a{t}^2}{2},y=at$

$\text{So,}$

$\text{2x}\text{=}\text{a+a}{\text{t}}^2,t=\frac{y}{a}$

$2x=a+a{\left(\frac{y}{a}\right)}^2$

$2x=a+a\frac{y^2}{a^2}$

$2x=a+\frac{y^2}{a}$

$2ax=a+\frac{y^2}{a}$

$2ax=a^2+y^2$

$y^2=2a(x-\frac{a}{2})$

It’s parabola with vertex at $(\frac{a}{2},0)$

Letus rectum $=2a.$

Hence, this is the answer.