Question

Two fixed parabolas with the same axis, focus of each being exterior to the other and the latus rectum 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. a parabola if $a=b$.
• C. a parabola if $a\ne b$.
• D. an ellipse if $b>a$.

Answer 1

Answer: (A), (C)

The equation of parabola can be taken as
$y^2=4a(x-K)$ and $y^2=-4b\left(x+K^{\prime }\right)$
A line parallel to the common axis is
$y=y_1$
then $A\left(\frac{y_1^2}{4a}+K,y_1\right)$ and $B\left(K^{\prime }-\frac{y_1^2}{4b},y_1\right)$
If $P\left(x_1,y_1\right)$ then
$2x_1=\frac{y^2}{4}\left[\frac{1}{a}-\frac{1}{b}\right]+K+K^{\prime }$ which is the required locus