By translating the axes the equation $x y - 2 x - 3 y - 4 = 0$ has changed to $X Y = k$ , then $k =$

- 10

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10

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4

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- 4

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Solution

The correct option is D $10$
Suppose the origin is shifted to $(α, β)$ .
Then $x - α = X$
and $y - β = Y$
Therefore $x = X + α$
$∴ y = Y + β$
Hence, $x y - 2 x - 3 y - 4 = 0$
$(X + α) (Y + β) - 2 (x + α) - 3 (Y + β) - 4 = 0$
$\Rightarrow X Y + β X + α Y + α β - 2 X - 2 α - 3 Y - 3 β - 4 = 0$
$\Rightarrow X Y + X (β - 2) + Y (α - 3) + α β - 2 α - 3 β - 4 = 0$
To make the linear terms disappear.
$\Rightarrow β - 2 = 0$
$\Rightarrow α - 3 = 0$
Hence, $(α, β) = (3, 2)$
Thus the axes have to be translated along $(3, 2)$ .
On substituting, we get
$X Y + 6 - 2 (3) - 3 (2) - 4 = 0$
$\Rightarrow X Y + 6 - 6 - 6 - 4 = 0$
$\Rightarrow X Y = 10$
Hence, $k = 10$

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