Question

A normal chord of the parabola subtending a right angle at the vertex makes an acute angle $\theta$ with the $x-$ axis, then $\theta -$

• A. $arc\tan 2$
• B. $arc\tan \sqrt{2}$
• C. $arc\cot \sqrt{2}$
• D. $noneofthese$

Answer 1

The correct option is B $arc\tan \sqrt{2}$

Led end points at chord is $(a{t}_1^2,2a{t}_1)$ and $(a{t}_2^2,2a{t}_2)$

Since it is normal to parabola $t_2=-t_1-\frac{-2}{t_1}$

equation of normal to parabola $y=-tx+2at+a{t}^3$

Slope is $-t_1$ which is given acute so $-t_1>0$ or $t_1<0$

now chord makes right angle at origin so

$\left\{\right(2a{t}_1-0\left)|\right(a{t}_1^2-0\left)\right\}\left\{\right(2a{t}_1-0\left)|\right(a{t}_1^2-0\left)\right\}=-1$

$t_1{t}_2=-4\to t_1(-t_1-2|t_1)=-t_1$

$t_1=-\sqrt{2}$

angle $\theta ={\tan }^{-1}\left(\sqrt{2}\right)$