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Shifting of Axes

On shifting t...

Question

On shifting the origin to a particular point, the equation $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ transforms to $X^{2} + Y^{2} = K$ . Then $K =$

12

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25

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24

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5

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Solution

The correct option is B $25$
Transformation of axis Let origin be shifted to (h, k), then
$(x, y) = (X - h, Y - k)$
$(X - h)^{2} + (Y - k)^{2} - 4 (X - h) - 6 (Y - k) - 12 = 0$
$X^{2} + Y^{2} - 2 X h - 4 X - 2 Y k - 6 Y + h^{2} + k^{2} + 4 h + 6 k - 12 = 0$
Comparing coefficients
$2 h = - 4$
$\Rightarrow h = - 2$
$2 k = - 6$
$k = - 3$
$∴ h^{2} + k^{2} + 4 h + 6 k - 12 = 4 + 9 - 8 - 18 - 12$
$= - 25$
$∴ X^{2} + Y^{2} = K = 25$

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