Question

Prove that the locus of the midpoints of the chords of contact of points on the director circle of the ellipse x²/a² + y²/b² = 1, with respect to the ellipse is (x²/a² + y²/b²)(a² + b²) = x² + y²

Answer 1

M(x₁, y₁) be the midpoint of the chord of contact of a point P(h, k) on the director circle x² + y² = a² + b².  Therefore

h² + k² = a² + b²  .....(1)

and  xx₁/a² + yy₁/b² = x²₁/a² + y₁²/b²  ....(2)

Also the chord of contact of P(h, k) with respect to the ellipse is

hx/a²  + ky/b² = 1   .....(3)

From Eqs. (2) and (3), we get

Substituting the values of h and k in Eq. (1), we have

Therefore, the locus of (x₁, y₁) is