Question

C is the centrec, AA' and BB' are major and minor axis of the ellipse. x²/a² + y²/b² = 1. If PN is the ordinate of a point P on the ellipse then show that (PN)²/(A'N)(AN) = (BC)²/(CA)²

Answer 1

Equation of the ellipse x²/a² + y²/b² = 1

P(a cos θ, b sin θ) any point on the ellipse. 

PN = b sin θ; AN = a – a cos θ, 

A'N = a + a cos θ ; BC = b, CA = a 

(A'N). (AN) = (a + a cos θ) (a – a cos θ) 

= a² – a² cos² θ. 

= a² (1 – cos² θ) = a² sin² θ