Question

A variable parabola of latus ractum $\phi$ touches a fixed equal parabola, then axes of the two curves being parallel. The locus of the vertex of the moving curve is a parabola, Whole latus rectum is

• A. $\phi$
• B. 2$\phi$
• C. 4$\phi$
• D. none of these

Answer 1

The correct option is B 2$\phi$

Let the parabola be $y^2=4ax..................\left(1\right)$ $(\phi =4a)$

Another Parabola with parallel axis and are equal and it touches $\left(1\right)$, Let it be, $(y-\beta {)}^2=-4a(x-\alpha )...............(2)$

As $\left(1\right)$ and $\left(2\right)$ touches each other,

$=>(y-\beta {)}^2=-4a(\frac{y^2}{4a}-\alpha )$

$=>2y^2-2\beta y+{\beta }^2-4a\alpha =0$

For equal roots $D=0$ as they touch each other,

$=>4{\beta }^2-8{\beta }^2+32a\alpha =0$

$=>-4{\beta }^2+32a\alpha =0$

where $(\alpha ,\beta )$ represents the vertex of the variable parabola.

$=>$locus of vertex$:4y^2=32ax$

$=>y^2=8ax$

$LR=8a=2\left(4a\right)$

$=2\phi .$$(as\phi =4a)$