A tangent at any point to the ellipse 4x2+9y2=36  is cut by the tangent at the extremities of the major axis at  T′ T  and '. The circle on   TT′ as diameter passes through the point.

Any point on the ellipse is  P(3cos⁡θ,2sin⁡θ)Equation of the tangent at P  is x3cos⁡θ+y2sin⁡θ=1 which  meets the tangents  x=3  and  x=−3 at the extremities of the major axis at T3,2(1−cos⁡θ)sin⁡θ  and T′−3,2(1+cos⁡θ)sin⁡θ Equation of the circle on  TT′  as diameter is     (x−3)(x+3)+y−2(1−cos⁡θ)sin⁡θy−2(1+cos⁡θ)sin⁡θ=0⇒x2+y2−4sin⁡θy−5=0  which passes through (5,0)