Question

Let there be two parabolas with the same axis, focus of each being exterior to the other and the latus recta being $4a$ and $4b$. The locus of the middle points of the intercepts between the parabolas made on the lines parallel to the common axis is a

• A. straight line if $a=b$
• B. parabola if $a\ne b$
• C. parabola for $a,b$
• D. none of these

Answer 1

Answer: (A), (B)

straight line if $a=b$
parabola if $a\ne b$

Let the equation of parabola are $y^2=-4bx+k_1$ and$y^2=4ax+k_2$
let $y=c$ is the line of equation parallel to the common axis.
Put $y=c$ in the parabola equation and find out the coordinates
The co-ordinates of intersection of a parabola with $y=c$
$(\frac{c^2-k_1}{-4b},c)$
The coordinates of intersection of the other parabola with $y=c$
$(\frac{c^2+k_1}{4a},c)$
$2h=\frac{c^2+k_2}{4a}+\frac{c^2-k_1}{-4b}2x=\frac{y^2}{4}(\frac{-1}{b}+\frac{1}{a})+\frac{k_2}{4b}-\frac{k_1}{4a}$
If $a=b$,It is a straight line
$a\ne b$, it is parabola.