Question

A circle passes through the point $A(1,0)$ and $B(5,0)$ and touches y-axis at $C(0,h)$. If $\angle ACB$ is maximum, then $h^2=$____

Answer 1

Let the $\angle ACB=\theta$ and the angle made by AC with Y-axis $=\alpha$;

$\tan \alpha =\frac{1}{h};\tan (\theta +\alpha )=\frac{5}{h}⟹\tan \theta =\frac{4h}{1+h^2}⟹0=\frac{4(1-h^2)}{(1+h^2{)}^2}⟹h^2=1$