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Shifting of Axes

The transform...

Question

The transformed equation of $x^{2} + 2 y^{2} + 2 x - 4 y + 2 = 0$ when origin is shifted to the point $(- 1, 1)$ with out rotation of axes, is

X^{2} - 2 Y^{2} = 1

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X^{2} + 2 Y^{2} = 1

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2 X^{2} + Y^{2} = 1

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X^{2} - Y^{2} = 0

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Solution

The correct option is B $X^{2} + 2 Y^{2} = 1$
Original equation is $x^{2} + 2 y^{2} + 2 x - 4 y + 2 = 0$
Origin is shifted to $(- 1, 1)$
$∴ x = X + h = X - 1 y = Y + k = Y + 1$
Transformed equation is
$(X - 1)^{2} + 2 (Y + 1)^{2} + 2 (X - 1) - 4 (Y + 1) + 2 = 0 \Rightarrow X^{2} - 2 X + 1 + 2 Y^{2} + 4 Y + 2 + 2 X - 2 - 4 Y - 4 + 2 = 0 \Rightarrow X^{2} + 2 Y^{2} = 1$

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