Question

Show that the locus of the midpoints of the chords of the ellipse x²/a² + y²/b² = 1 which subtend right angle at the centre of the ellipse is a² + b²/a²b²(x²/a² + y²/b²)² = x²/a⁴ + y²/b⁴

Answer 1

Let PQ be a chord whose midpoint is (x₁, y₁) which is subtending right angle at the centre. Hence, the equation of the chord PQ is

Now, the combined equation of the pair of lines CP and CQ is

Now ∠PCQ 90° which implies that in the above equation

Coefficient of x² + Coefficient of y² = 0

Hence, the locus of (x₁, y₁) is