Let x^2 f(a^2+7a+3) + y^2 f(3a+15) = 1 represent an ellipse with major axis along y -axis, where f is a strictly decreasing positive function on R . If the set of all possible values of a is R - [ , ] , then ^2+ ^2 is equal to:

Step-by-step solution

For the given equation to represent an ellipse with its major axis along the y -axis, the denominator of y^2 must be strictly greater than the denominator of x^2 . f(3a+15) > f(a^2+7a+3) Since f is given as a strictly decreasing positive function on R , the inequality sign reverses for the arguments: 3a+15 a^2 + 4a - 12 > 0 (a+6)(a-2) > 0 The solution to this inequality is a (- , -6) (2, ) . This can be rewritten in terms of the set difference as a R - [-6, 2] . Comparing this with the given set R - [ , ] , we get