Question

A tangent to an ellipse x²/a² + y²/b² = 1 is cut by the tangents at the vertices T and T a. Prove that the circle with T and T' as extremities of a diameter passes through the foci of the ellipse.

Answer 1

The tangent at P(a cos θ, b sin θ) is 

 cosθ/a + ysinθ/b = 1

The tangents at A(a, 0) and A'(−a, 0) are x = a and x =−a, respectively. Therefore

Hence, the circle with T and T' as the ends of a diameter is

which passes through S(ae, 0) and Sa(−ae, 0).