Please help with question beginning with "SHOW".

Let Q (X) = X . M . X be a strictly positive-definite quadratic form on R, that is XE R and M is a symmetric, positive-definite, 2 x 2 real matrix, of Tr(M) = T and determinant det(M) = D, with T, D >0. If Q(X) = ax, + 2bX1X2 = cX2, a > c and the angle 0 = = tan-1 . 2b - for which 2 a the rotation of Cartesian coordinate axes by 0 brings M to its diagonal form, or YTR(-0)MR(0)Y = T2 T2 + DY ? + NIN 4 D ) Y2, where R(0) is the rotation matrix of angle 0. Show that the transformations bringing Q(Vx) to its canonical form, the Laplace operator in the new coordinates, consist of the rotation by angle 0 described above followed by dilation of coordinates by the factors 1+, where 14 = T2 D 2 4