In order to eliminate the first degree terms from the equation $4 x^{2} + 8 x y + 10 y^{2} - 8 x - 44 y + 14 = 0$ the point to which the origin has to be shifted is

(- 2, 3)

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(2, - 3)

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(1, - 3)

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(- 1, 3)

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Solution

The correct option is A $(- 2, 3)$
A two-degree curve is always symmetric about its centre. Therefore, the origin should be shifted to the centre of this curve.

To calculate the centre, we partially differentiate the curve
$f (x, y) = 4 x^{2} + 8 x y + 10 y^{2} - 8 x - 44 y + 14 = 0$

$\frac{\partial f}{\partial x} = 8 x + 8 y - 8$
$\frac{\partial f}{\partial y} = 8 x + 20 y - 44$

Thus we have the equation:
$8 x + 8 y = 8$ and $8 x + 20 y = 44$

By solving the above two equations, we get the centre as $x = - 2$ and $y = 3$

Thus the centre lies at $(- 2, 3)$

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