Director's circle of ellipse and the diametric circle of any focal chord.

Question

Problem: Prove that a circle drawn with any focal chord of an ellipse touches it's director circle i.e the locus of intersection of perpendicular tangents to the ellipse

I need to prove that the circle with any focal chord as diameter of a standard ellipse $x^2/a^2 + y^2/b^2 = 1$ touches the director's circle: $x^2 + y^2 = a^2 + b^2$.

I have reached the result well using analytical geometry, but I am finding a method using pure geometry and having some trouble with that. I tried to use some geometrical propositions of conics, but I didn't reach anywhere.

Answer 1

As stated, the theorem you want to prove is false. See figure below, where $AB$ is a focal chord and the black circle is the director circle of the ellipse.

To give neat counterexample, take as focal chord the major axis: the circle having it as diameter is centred at the origin and its radius is $a$, hence it is inside the director circle.

Answer 2

This is not an answer.

I agree with Aretino. This theorem has no validity. In the diagram given below one can see that the green circles fail to touch the black circle. As the length of the focal chord increases, the gap between the 2 circles monotonically widened. How did you (I mean OP) reach the result using analytical geometry?