A tangent at any point to the ellipse 

$4x^2+9y^2=36$ is cut by the tangent at the extremities of the 

major axis at $T^{\mathrm{\prime }}$ $T$ and '. The circle on   $T{T}^{\mathrm{\prime }}$ as diameter passes 

through the point.  

Any point on the ellipse is  $P(3\cos \theta ,2\sin \theta )$

Equation of the tangent at $P$  is $\frac{x}{3}\cos \theta +\frac{y}{2}\sin \theta =1$ which  

meets the tangents  $x=3$ and $x=-3$ at the extremities of

 the major axis at 

$T\left(3,\frac{2(1-\cos \theta )}{\sin \theta }\right)$ and $T^{\mathrm{\prime }}\left(-3,\frac{2(1+\cos \theta )}{\sin \theta }\right)$ 

Equation of the circle on  $T{T}^{\mathrm{\prime }}$  as diameter is 

    $(x-3)(x+3)+\left(y-\frac{2(1-\cos \theta )}{\sin \theta }\right)\left(y-\frac{2(1+\cos \theta )}{\sin \theta }\right)=0$

$\Rightarrow x^2+y^2-\frac{4}{\sin \theta }y-5=0$  which passes through $(\sqrt{5},0)$