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Q.

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

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a

$x^{2} + 2 y^{2} = 1$

b

$2 x^{2} + y^{2} = 1$

c

$x^{2} + y^{2} = 1$

d

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

answer is B.

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Detailed Solution

$θ = 45^{0}$ Given equation is $3 x^{2} + 3 y^{2} + 2 xy = 2$ T.E is
$3 [{(\frac{x - y}{\sqrt{2}})}^{2} + {(\frac{x + y}{\sqrt{2}})}^{2}] + 2 (\frac{x - y}{\sqrt{2}}) (\frac{x + y}{\sqrt{2}}) = 2$
$3 (x^{2} + y^{2}) + x^{2} - y^{2} = 2$
$4 x^{2} + 2 y^{2} = 2$
$2 x^{2} + y^{2} = 1$

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The transformed equation of 3x2+3y2+2xy=2 when the coordinate axes are rotated through an angle of 45°