When the origin is shifted to a suitable point, the equation $2 x^{2} + y^{2} - 4 x + 4 y = 0$ transforms as $2 X^{2} + Y^{2} - 8 X + 8 Y + 18 = 0$ . The point to which origin was shifted is

(1, 2)

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(1, - 2)

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(- 1, 2)

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(- 1, - 2)

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Solution

The correct option is C $(- 1, 2)$
Let the origin be shifted to $(α, β)$ .
Then
$x - α = X$
$y - β = Y$
Therefore
$x = X + α$
$y = Y + β$
$2 x^{2} + y^{2} - 4 x + 4 y = 0$ transforms to
$2 (X + α)^{2} + (Y + β)^{2} - 4 (X + α) + 4 (Y + β) = 0$
$2 X^{2} + Y^{2} + 2 β Y + 4 α X + 2 α^{2} + β^{2} - 4 X - 4 α + 4 Y + 4 β = 0$

$2 X^{2} + Y^{2} + X (4 α - 4) + Y (2 β + 4) + 2 α^{2} + β^{2} - 4 α + 4 β = 0$
Comparing with,
$2 X^{2} + Y^{2} - 8 X + 8 Y + 18 = 0$

$4 α - 4 = - 8$
$4 α = - 4$
$α = - 1$ and
$2 β + 4 = 8$
$2 β = 4$
$β = 2$
Thus
$(α, β) = (- 1, 2)$ .

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