Mathematics - Parabola Question with Solution | TestHub

Any point on circle $x^2+y^2=4$ is $(2\cos \alpha ,2\sin \alpha )$

∴ Equation of directrix is $x(\cos \alpha )+y(\sin \alpha )-2=0$

Let focus be $\left(x_1,y_1\right)$. Then as $A(1,0),B(-1,0)$ lie on parabola we must have $\begin{array}{l}{\left(x_1-1\right)}^2+y_1^2=(\cos \alpha -2{)}^2{\left(x_1+1\right)}^2+y_1^2-(\cos \alpha +2{)}^2\end{array}\}\Rightarrow x_1=2\cos \alpha ,y=\pm \sqrt{3}\sin \alpha$

∴ Length of semi latus-rectum of parabola $={\perp }^r$ distance from focus to directrix $|2\pm \sqrt{3}|{\sin }^2\alpha$ Hence, maximum possible length $=2-\sqrt{3}$