Point P is on the circle x^2 + y^2 = a^2. AB is the chord of contact of P with respect to the ellipse

Question

Point P is on the circle x2 + y2 = a2. (bar)AB is the chord of contact of P with respect to the ellipse x2/a2 + y2/b2 = 1

Then, show that the locus of midpoint of (bar)AB is curve

(x2/a2 + y2/b2)2 = x2 + y2/a2

Answer 1

Let P(a cos θ, a sinθ) be a point on x2 + y2 + a2. Then, the chord of contact AB of P with respect to the ellipse x2/a2 + y2/b2 = 1

As in the case of circle and parabola, we can see that the equation of the chord of contact of P(x1, y1) with respect to the ellipse is

Suppose M(x1, y1) is the midpoint of (bar) AB so that the equation of chord (bar)AB is

From Eqs. (1) and (2)

Therefore, the locus of M(x1, y1) is