Let the point P(4, 1) lie on the ellipse x^2 a^2 + y^2 b^2 = 1 , where a > b . If the absolute difference of the focal distances of the point P is 4 , then the product of the focal distances of P is

Step-by-step solution

Let the focal distances of the point P(x₁, y₁) on the ellipse be SP and S'P . We know that SP = a - ex₁ and S'P = a + ex₁ . The absolute difference of the focal distances is |S'P - SP| = 2e|x₁| . Given that the absolute difference is 4 and x₁ = 4 , we have: 2e(4) = 4 8e = 4 e = 1 2 . Since P(4, 1) lies on the ellipse, it satisfies the equation: 16 a^2 + 1 b^2 = 1 Using the relation b^2 = a^2(1 - e^2) , we get: b^2 = a^2 (1 - 1 4 ) = 3a^2 4 Substituting b^2 into the ellipse equation: 16 a^2 + 4 3a^2 = 1 48 + 4 3a^2