Mathematics - Parabola Question with Solution | TestHub

Equation of Tangent is $yt=x+a{t}^2$

$\therefore \mathrm{Tan}\alpha =\frac{1}{t};t=\cot \alpha$

Equation of parallel normal is $yt=x+K$ $\begin{array}{rlr}& a\cdot 1^3+2a\cdot 1\cdot (-t{)}^2+(-t{)}^2\cdot K=0& K=\frac{-\left(a+2a{t}^2\right)}{t^2}\end{array}$

Distance $=\frac{a{t}^2+\frac{a+2a{t}^2}{t^2}}{\sqrt{1+t^2}}=\frac{a{t}^4+2a{t}^2+a}{t^2\sqrt{1+t^2}}=\frac{a{\left(t^2+1\right)}^{3/2}}{t^2}$