GATE 2016 EE SET 2 – Question 32

Given: $$P=\begin{bmatrix}3 & 1\\ 1 & 3\end{bmatrix}$$

$$\binom{a}{b}=P\binom{x}{y}$$

$$\begin{bmatrix}3 & 1\\ 1 & 3\end{bmatrix}\binom{x}{y}=\binom{a}{b}$$

$$3x+y=a ; x+3y=b ; a^{2}+b^{2}=1$$

$$(3x+y)^{2}+(x+3y)^{2}=1$$

$$9x^{2}+y^{2}+6xy+x^{2}+9y^{2}+6xy=1$$

$$10x^{2}+10y^{2}+12xy=1$$

$$8(x+y)^{2}+2(x-y)^{2}=1$$

$$\frac{(x+y)^{2}}{\left ( \frac{1}{8}\right )}+\frac{(x-y)^{2}}{\left ( \frac{1}{2} \right )}=1$$
Major axis length $$a< b \Rightarrow x+y=0 \Rightarrow x=-y$$