The maximum and minimum values of the product of the focal distances of a point on the ellipse x^2 a^2 + y^2 b^2 = 1 ( a > b ) are 25 and 16 respectively. A tangent to this ellipse in the first quadrant forms a triangle of minimum area with the coordinate axes. The radius of the circumcircle of this triangle is

Step-by-step solution

The product of the focal distances of a point P(x, y) on the ellipse is given by SP S'P = a^2 - e^2 x^2 . Since -a x a , the maximum value occurs at x = 0 , giving a^2 = 25 a = 5 . The minimum value occurs at x = a , giving a^2 - e^2 a^2 = a^2(1 - e^2) = b^2 . Thus, b^2 = 16 b = 4 . The equation of the tangent at a point (a , b ) in the first quadrant is: x 5 + y 4 = 1 The intercepts on the coordinate axes are X = 5 and Y = 4 . The area of the triangle formed with the coordinate axes is: = 1 2 X Y = 1 2 ( 5 ) ( 4 )