Question

Let $A$ be the vertex of the parabola $y^2=4ax$ and the length of focal chord $PQ$ is $l\text{units}$. If perpendicular distance from vertex to $PQ$ is $p\text{units}$, then

• A. $l$ is proportional to $p^2$
• B. $l$ is inversely proportional to $p^2$
• C. $l$ is proportional to $p$
• D. $l$ is inversely proportional to $p$

Answer 1

The correct option is B $l$ is inversely proportional to $p^2$

Given parabola is $y^2=4ax$
Vertex $=(0,0)$
As $PQ$ is focal chord, so
$l=a{(t+\frac{1}{t})}^2\cdots \left(1\right)$
Also, equation of the chord is
$2x-y(t_1+t_2)+2a{t}_1{t}_2=0$

Equation of focal chord $PQ$ is
$2x-y(t-\frac{1}{t})-2a=0$
Distance from vertex is
$p=\frac{|2a|}{\sqrt{{(t-\frac{1}{t})}^2+4}}\Rightarrow p^2=\frac{4a^2}{{(t+\frac{1}{t})}^2}$
Using equation $\left(1\right)$, we get
$l{p}^2=4a^3\left(\text{constant}\right)$