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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

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