 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/13/b/i/solution

Multiply the quadratic form out directly:

(x y) A (x y) = x (5 x - 3 y) + y (- 3 x + 5 y) = 5 x^{2} - 6 x y + 5 y^{2} .

Thus $b = 5$ , $c = - 6$ and $d = 5$ . Both eigenvalues are positive, so this is a positive-definite quadratic form; its positive unit level set is an ellipse. The next part exhibits its principal axes explicitly.

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