Question

If $p$ be the perpendicular distance of a focal chord $PQ$ of length $l$ from the vertex $A$ of the parabola $y^2=4ax$, then $l$ varies inversely as

• A. $p^2$
• B. $\frac{1}{p^2}$
• C. $p$
• D. $\frac{1}{p}$

Answer 1

The correct option is B $\frac{1}{p^2}$

$\because$ $PQ$ is focal chord
$\therefore l=a{(t+\frac{1}{t})}^2$
Also chord equation joining points $t_1,t_2$ is
$y(t_1+t_2)-2x-2a{t}_1{t}_2=0$
$\because$ $PQ$ is focal chord
$y(t-\frac{1}{t})-2x+2a=0$
$p=\frac{2a}{\sqrt{{(t-\frac{1}{t})}^2+4}}$
$\therefore p^2=\frac{4a^2}{{(t+\frac{1}{t})}^2}$
Clearly, $l{p}^2=4a^3$(constant)