this question and solution about quadratic surfaces:

Describe the quadric surface given by the equation 4x' + 913 + 523 - 4xy + Byz + 12x2 + 92 = 3. Solution Write the equation as The eigenvalues of the matrix 4 -2 6 M = -2 9 4 6 4 un are 9, 12 and -3, and one finds that E4 = E12 = NN J E-3 = WIN NL It follows that 2 1 P = -2 2 -1 12 2 is an orthogonal matrix such that PIMP = diag (9,12, - 3). We find that det P = 1, so that PE SO (3,R). Writing = P (1) Yields (ry,2)PT 4 -26 P/1" + (0,0,9)P/x) - 3=0, -294 6 y' 4 5 177 7.2 Quadric surfaces Le. 3(1)+4() - (=)+1+ 2+2:'-1=0 (after division by 3). Completing the square, we find that and setting Perth Wyth "=2-1 (2) finally yields = 1. This is the equation of a hyperboloid of one sheet. . The change of variable (1) is a rotation through the angle # = -+ about an axis in the direction n = . The change of variable (2) is a translation