If P is any point on the ellipse a 2 x 2 + b 2 y 2 = 1 and S is the focus of the ellipse and A 2 A 1 (where A 1 & A 2 are co-prime natural numbers) is ratio of the area of the ellipse an

Question

If P is any point on the ellipse a 2 x 2 ​ + b 2 y 2 ​ = 1 and S is the focus of the ellipse and A 2 ​ A 1 ​ ​ (where A 1 ​ & A 2 ​ are co-prime natural numbers) is ratio of the area of the ellipse and the area enclosed by the locus of mid-point of PS, value of A 2 ​ A 1 ​ ​ is ____

TestHub · Accepted Answer

Let a 2 x 2 ​ + b 2 y 2 ​ = 1. be an ellipse\ Area = A 1 ​ = π ab Let ( h , k ) be the mid point of PS ⇒ 2 h = a cos θ + ae &2 k = b sin θ Eliminating θ we get ( 2 a ​ ) 2 ( x − 2 ae ​ ) 2 ​ + ( 2 b ​ ) 2 y 2 ​ = 1. A 2 ​ = π 2 a ​ 2 b ​ = 4 π ab ​ . ⇒ A 1 ​ : A 2 ​ = 4 : 1 y x ​ = 4 .

Mathematics - Ellipse Question with Solution | TestHub

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JEE Advanced (2028)