Let a, b, c R, and let f (x1, x2) = ax2 1 + 2bx1x2 + cx2 2 . a. The Spectral Theorem

QUESTION:

Let a, b, c R, and let f (x1, x2) = ax2 1 + 2bx1x2 + cx2 2 . a. The Spectral Theorem tells us that there exists an orthonormal basis for R2 with respect to whose coordinates (y1, y2) we have f (x1, x2) = f (y1, y2) = y2 1 + y2 2 . Show that the y1y2-axes are obtained by rotating the x1x2-axes through an angle , where cot 2 = a c 2b . Determine the type (ellipse, hyperbola, etc.) of the conic section f (x1, x2) = 1 from a, b, and c. (Hint: Use the characteristic polynomial to eliminate 2 in your computation of tan 2.) b. Use the formula for f above to find the maximum and minimum of f (x1, x2) on the unit circle x2 1 + x2 2 = 1.