The transformed equation of 5x2+4xy+8y2-12x-12y when axes are translated to the point 1,12 is

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The transformed equation of $5 x^{2} + 4 xy + 8 y^{2} - 12 x - 12 y$ when axes are translated to the point $(1, \frac{1}{2})$ is

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$5 x^{2} + 4 xy + 8 y^{2} = 6$

$5 x^{2} + 4 xy + 8 y^{2} = 3$

$5 x^{2} + 4 xy + 8 y^{2} = 12$

$5 x^{2} + 4 xy + 8 y^{2} = 9$

answer is C.

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Detailed Solution

$Given origin shifted to the point = (1, \frac{1}{2}) Now x = X + 1, y = Y + \frac{1}{2} Given original equation 5 x^{2} + 4 xy + 8 y^{2} - 12 x - 12 y = 0 Now transformed equation is 5 {(X + 1)}^{2} + 4 (X + 1) (Y + \frac{1}{2}) + 8 {(Y + \frac{1}{2})}^{2} - 12 (X + 1) - 12 (Y + \frac{1}{2}) = 0 \Rightarrow 5 (X^{2} + 1 + 2 X) + 2 (X + 1) (2 Y + 1) + 2 {(2 Y + 1)}^{2} - 12 (X + 1) - 6 (2 Y + 1) = 0 \Rightarrow 5 X^{2} + 5 + 10 X + 4 X Y + 2 X + 4 Y + 2 + 8 Y^{2} + 2 + 8 Y - 12 X - 12 - 12 Y - 6 = 0 \Rightarrow 5 X^{2} + 4 X Y + 8 Y^{2} - 9 = 0$