GATE 2016 EE (EE2) – Question 42
Let
$$P=\begin{bmatrix}3&1\\1&3\end{bmatrix}.$$
The set S consists of vectors (x,y) such that $a^2+b^2=1$, where $(a,b)^T=P(x,y)^T$. Then S is
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Correct answer: (D) an ellipse with minor axis along $(1,1)^T$
Explanation
P has orthogonal eigenvectors (1,1) and (1,−1), with eigenvalues 4 and 2. In normalized coordinates along these directions, the condition |Px|²=1 becomes 16u²+4v²=1.
The semiaxes are 1/4 in the (1,1) direction and 1/2 in the (1,−1) direction. Thus the shorter, minor axis is along **(1,1), D**.