Question

If a chord which is normal to the parabola at one end and subtends a right angle at the vertex, then slope of the chord is

• A. 1
• B. $-2$
• C. $\sqrt{2}$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

Answer: (C)

Chord $PQ$ is normal to the parabola at Qand is equation is $y=mx-2am-a{m}^3$

The slope of normal $PQ$ is $m=\tan \theta .$ The joint equation of $OP$ and $OQ$ is obtained by making the equation of parabola $y^2=4$ ax homogeneous with the help of $(1)$ Thus joint equation of lines $OP,OQ$ is
$y^2=4ax\left\{\frac{mx-y}{2am+a{m}^3}\right\}$
or $m\left(2+m^2\right)y^2+4xy-4m{x}^2=0$
Since $\angle POQ=\frac{\pi }{2}$, the sum of the coefficients of $x^2$ and $y^2$ in $(2)$ must be zero.
$m\left(2+m^2\right)-4m=0$ or $m\left(m^2-2\right)=0$
As $m\ne 0$, we get $m^2=2$ or $m=\pm \sqrt{2}$. Thus, $m=\sqrt{2}$.