Let A = bmatrix x & 1-x^2 1-y^2 & y bmatrix be a matrix where x, y [-1, 1] . If the determinant of A is 1 2 , and M and m denote the maximum and minimum values of x^2 + y^2 respectively, then the value of 8(M + m) is

Step-by-step solution

The determinant of matrix A is given by: (A) = xy - 1-x^2 1-y^2 = 1 2 Let u = ⁻¹x and v = ⁻¹y , where u, v [0, ] . Then x = u and y = v . Since u, v [0, ] , we have 1-x^2 = u and 1-y^2 = v . Substituting these into the determinant equation: u v - u v = 1 2 (u+v) = 1 2 Since u, v [0, ] , u+v [0, 2 ] . Thus, u+v = 3 or u+v = 5 3 . We need to find the extrema of x^2 + y^2 = ^2 u + ^2 v . Using trigonometric identities: ^2 u + ^2 v = 1 + 2u + 2v 2 = 1 + (u+v) (u-v) Since (u+v) = 1 2 , the expression simplifies to: x^2