Question

Two mutually perpendicular tangents of the parabola $y^2=4ax$ at the points $Q_1$ and $Q_2$ on it meet its axis in $P_1$ and $P_2.$ If $S$ is the focus of the parabola, then the value of ${\left(\frac{1}{S{P}_1}+\frac{1}{S{P}_2}\right)}^{-1}$ is equal to

• A. $\frac{a}{4}$
• B. $\frac{a}{2}$
• C. $a$
• D. $2a$

Answer 1

Answer: (C)

Let $Q_1=\left(a{t}_1^2,2a{t}_1\right)$ & $Q_2=\left(a{t}_2^2,2a{t}_2\right)$ are points of contact of the tangents passing through $P_1$ & $P_2$ respectively.
$\Rightarrow P_1=\left(-a{t}_1^2,0\right)$ & $P_2=\left(-a{t}_2^2,0\right)$
$\Rightarrow S{P}_1=a+a{t}_1^2$ & $S{P}_2=a+a{t}_2^2$
$\Rightarrow \frac{1}{S{P}_1}+\frac{1}{S{P}_2}=\frac{1}{a+a{t}_1^2}+\frac{1}{a+a{t}_2^2}=\frac{1}{a+a{t}_1^2}+\frac{1}{a+a{\left(-\frac{1}{t_1}\right)}^2}$
$=\frac{1}{a}$
$\therefore {\left(\frac{1}{S{P}_1}+\frac{1}{S{P}_2}\right)}^{-1}=a$