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Chords at right angles are drawn through...

Chords at right angles are drawn through any point P of the ellipse, and the line joining their extremities meets the normal in the point Q. Prove that Q is the same for all such chords, its coordinates being,`(a^3e^2cos alpha)/(a^2+b^2)` and `(-a^2be^2sin alpha)/(a^2+b^2)`. Prove also that the major axis is the bisector of angle PCQ and that the locus of Q for different positions of P is the ellipse `x^2/a^2 + y^2/b^2 = ((a^2-b^2)/(a^2+b^2))^2`

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If the chords be PK and PK' let the equation to KK' be y = mx + c; transform the origin to P and by means of ART 12, find the condition that the angle KPK' is a right angle ; substitute for c in the equation to KK' and find the point of inter section of KK' and the normal at P.
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