If p and p ' denote the lengths of the perpendicular from a focus and the centre of an ellipse with semi-major axis of length a , respectively, on a tangent to the ellipse and r denotes the focal distance of the point, then

Step-by-step solution

Let the ellipse be x 2 a 2 + y 2 b 2 = 1 and let x cos θ a + y sin θ b = 1 . . . ( i ) be a tangent to it at point ( a cos θ , b sin θ ) . Then, p = length of the perpendicular from S ( a e ,   0 ) on Eq. ( i ) p = e cos θ - 1 cos 2 θ a 2 + sin 2 θ b 2 p ' = length of the perpendicular from O ( 0 , 0 ) on Eq. ( i ) ⇒   p ' = 1 cos 2 θ a 2 + sin 2 θ b 2 and r = a e cos θ - a Clearly, r p ' = a p