 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/6/ii/solution

Use the orthogonal change of coordinates

x^{'} = O^{T} x, x = O x^{'} .

It preserves distances and angles and gives

x^{T} M x = x^{' T} D x^{'}

. Thus the quadric surface is expressed along its principal axes as

λ_{1} x_{1}^{'2} + λ_{2} x_{2}^{'2} + λ_{3} x_{3}^{'2} = k .

The following three cases are the subparts of (ii) in the original PDF.

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