Prove that the locus of the middle point...

Prove that the locus of the middle points of the chords of contact of the tangents drawn from any point on the circle `x^2 + y^2 = c^2` to the hyperbola `x^2/a^2 - y^2/b^2 = 1` is `(x^2/a^2 - y^2/b^2)^2 = (x^2 + y^2)/(c^2)`.

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