Mathematics - Parabola Question with Solution | TestHub

The normal at $P\left(a{t}_1{}^2,2a{t}_1\right)$ is $y+x{t}_1=2a{t}_1+a{t}_1{}^3$ with slope say $\tan \alpha =-t_1=\frac{1}{\sqrt{3}}$. If it meets curve at $Q\left(a{t}_2{}^2,2a{t}_2\right)$ then $t_2=-t_1-\frac{2}{t_1}=\frac{7}{\sqrt{3}}$. Then angle $\theta$ between parabola (tangent at Q ) and normal at P is given by $\tan \theta =\frac{-t_1-\frac{1}{t_2}}{1-\frac{t_1}{t_2}}=\frac{1}{2\sqrt{3}}$

$\Rightarrow \theta ={\tan }^{-1}\left(\frac{1}{2\sqrt{3}}\right)$