Question

Consider a parabola $y^2=4ax$, the length of focal chord is $l$ and the length of the perpendicular from vertex to the chord is $p$ then

• A. $l.p$ is constant
• B. $l.p^2$ is constant
• C. $l^2.p$ is constant
• D. none of these

Answer 1

Answer: (B)

$l.p^2$ is constant

Let $P(a{t}^2,2at)$ and $Q(a/t^2,-2a/t)$ be a focal chord of the parabola (as $t_1{t}_2=-1$)
Length of PQ$=l=\sqrt{{(a{t}^2-a/t^2)}^2+{(2at+2a/t)}^2}$
$=a\sqrt{{(t^2-1/t^2)}^2+4{(t+1/t)}^2}$
$=a(t+1/t)\sqrt{{(t-1/t)}^2+4}$
$=a{(t+1/t)}^2$

Length of the perpendicular from the vertex (0, 0) on the line PQ whose equation is
$y(t-\frac{1}{t})=2(x-a)$ is given by
$P=\frac{2a}{\sqrt{{(t-1/t)}^2+2^2}}=\frac{2a}{(t+1/t)}$

so that $l.p^2=\frac{4a^2}{{(t+1/t)}^2}\times a{(t+1/t)}^2=4a^3$ which is constant.