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The transformed equation of $2 x^{2} + 3 y^{2} + 8 x - 18 y + 6 = 0$ when the axes are translated to the point $(- 2, 3)$ is

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$3 X^{2} - 2 Y^{2} = 23$

$3 X^{2} + 2 Y^{2} = 23$

$2 X^{2} - 3 Y^{2} = 29$

$2 X^{2} + 3 Y^{2} = 29$

answer is B.

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Detailed Solution

$(h, k) = (- 2, 3)$
$x = X - 2; y = Y + 3$
Transformed equation is
$2 {(X - 2)}^{2} + 3 {(Y + 3)}^{2} + 8 (X - 2) - 18 (Y + 3) + 6 = 0$
$\Rightarrow 2 X^{2} + 3 Y^{2} = 29$

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