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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.

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    $\begingroup$ Please show your solution using analytical geometry. Someone may be able to see key "pure" geometrical properties in the equations and/or your process, without having to waste time deriving those equations themselves. $\endgroup$
    – Blue
    Commented Jan 17, 2020 at 12:41
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    $\begingroup$ As noted, the theorem is false. The "focal diameter circles" are not typically tangent to the director circle. They are, however, tangent to a conic's "associate circles". This is the topic of the question "an important property of an ellipse" from back in 2011. Answers include some animations. $\endgroup$
    – Blue
    Commented Jan 19, 2020 at 12:54

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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?

enter image description here

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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.

enter image description here

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    $\begingroup$ @ChiefVS Just take a particular case, for instance take as focal chord the major axis: the circle having that as diameter is centred at the origin and its radius is $a$, hence it cannot touch the director circle. $\endgroup$
    – Intelligenti pauca
    Commented Jan 18, 2020 at 16:21

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