Mathematics - Parabola Question with Solution | TestHub

Normal at (at ${}^2$, 2at) is $\mathrm{y}+\mathrm{tx}=2\mathrm{at}+{\mathrm{at}}^3\left(\mathrm{a}=\frac{1}{4}\right)$

if this passes through $(\mathrm{c},0)$, we have $\begin{array}{rlrl}& ct=2at+a{t}^3=\frac{t}{2}+\frac{t^3}{4}& \Rightarrow t\left[t^2+2-4c\right]=0& \Rightarrow t=0\text{ or }{t}^2=4c-2\end{array}$

if $\mathrm{t}=0$ the point at which the normal is drawn is $(0,0)$.

if $\mathrm{t}\ne 0$ then the two values of t represents slope of normals through $(\mathrm{c},0)$.

if these normals are perpendicular then $\left(-{\mathrm{t}}_1\right)\left(-{\mathrm{t}}_2\right)=-1$ $\begin{array}{rlrl}& \Rightarrow {\mathrm{t}}_1{\mathrm{t}}_2=-1& \Rightarrow (\sqrt{4\mathrm{c}-2})(-\sqrt{4\mathrm{c}-2})=-1& \Rightarrow \mathrm{c}=\frac{3}{4}\end{array}$