Mathematics - Parabola Question with Solution | TestHub

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

equation of the directrix is $\pi (\cos \alpha )+y(\sin \alpha )-2=0$.

Let the focus be $(x_1,y_1)$. Then, as $A(1,0)$ and $B(-1,0)$ lie on the parabola, we must have:

$$\begin{array}{l}(x_1-1{)}^2+y_1^2=(\cos \alpha -2{)}^2\\ (x_1+1{)}^2+y_1^2=(\cos \alpha +2{)}^2\end{array}\}\Rightarrow x_1=2\cos \alpha ,y_1=\pm \sqrt{3}\sin \alpha$$

length of the semi-latus rectum of the parabola is the perpendicular distance from the focus to the directrix:

$$|2\pm \sqrt{3}|{\sin }^2\alpha$$

Hence, the maximum possible length is $2+\sqrt{3}$