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Q.

The transformed equation of $3 x^{2} + 3 y^{2} + 2 x y = 2$ when the coordinate axes are rotated through an angle of $45^{0}$ is

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a

$X^{2} + Y^{2} = 1$

b

$2 X^{2} + Y^{2} = 1$

c

$X^{2} + 2 Y^{2} = 1$

d

$X^{2} + 3 Y^{2} = 1$

answer is B.

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Detailed Solution

$\begin{array}{l} x = X \cos θ - Y \sin θ, y = X \sin θ + Y \cos θ \\ \Rightarrow x = \frac{X - Y}{\sqrt{2}} \\ \Rightarrow y = x \sin 45^{0} + y \cos 45^{0} = \frac{X + Y}{\sqrt{2}} \\ S u b s t i t u t i n g i n 3 x^{2} + 3 y^{2} + 2 x y = 2 \end{array}$

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