Let P be the point to which origin has to be shifted by the translation of axes, so as to remove the first degree terms from the equation $3 x^{2} + y^{2} - 6 x + 4 y + 4 = 0$ . If the origin is shifted to P by the translation of axes, then the transformed equation of $2 x^{2} + 3 x y - 5 y^{2} + 2 x - 23 y - 24 = 0$ is

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$2 x^{2} + 3 x y - 5 y^{2} - 13 = 0$

$2 x^{2} - 3 x y + 5 y^{2} = 0$

$2 x^{2} + 3 x y - 5 y^{2} = 0$

$x^{2} + 4 x y - 3 y^{2} - 4 x + 20 y + 23 = 0$

answer is C.

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Detailed Solution

New origin $= (\frac{- g}{a}, \frac{- f}{b}) = (1, - 2) = (h, k)$
$\begin{aligned} x = X + h = X + 1 \\ y = Y + k = Y - 2 \end{aligned}$
$∴$ The transformed equation is $2 x^{2} + 3 x y - 5 y^{2} = 0$

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