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

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

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

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

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

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Solution

The correct option is B $2 X^{2} + Y^{2} = 1$
When axes are rotated through $45^{0}$ ,
$x = x c o s θ - y s i n θ = \frac{x - y}{\sqrt{2}}$
$y = x s i n θ - y c o s θ = \frac{x + y}{\sqrt{2}}$
$3 x^{2} + 3 y^{2} + 2 x y = 2 \Rightarrow 3 (\frac{x - y}{\sqrt{2}})^{2} + 3 (\frac{x + y}{\sqrt{2}})^{2} + 2 (\frac{x^{2} - y^{2}}{2}) = 2$
$3 (x^{2} + y^{2} - 2 x y) + 3 (x^{2} + y^{2} + 2 x y) + 2 x^{2} - 2 y^{2} = 4$
$4 (2 x^{2} + y^{2}) = 4$
$\Rightarrow 2 x^{2} + y^{2} = 1$

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