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(D) Let the origin be shifted to the point $P(h, k)$.Then,the transformation equations are $x = X+h$ and $y = Y+k$.Substituting these into the given e

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$x+2y+5=0$

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(D) Let the origin be shifted to the point $P(h, k)$.Then,the transformation equations are $x = X+h$ and $y = Y+k$.Substituting these into the given equation $2x^2+y^2-4x+4y=0$:$2(X+h)^2 + (Y+k)^2 - 4(X+h) + 4(Y+k) = 0$$2(X^2+2hX+h^2) + (Y^2+2kY+k^2) - 4X - 4h + 4Y + 4k = 0$$2X^2 + Y^2 + (4h-4)X + (2k+4)Y + (2h^2+k^2-4h+4k) = 0$.Comparing this with the transformed equation $2X^2+Y^2-8X+8Y+18=0$:$4h-4 = -8$ $\Rightarrow 4h = -4$ $\Rightarrow h = -1$.$2k+4 = 8$ $\Rightarrow 2k = 4$ $\Rightarrow k = 2$.Thus,the origin is shifted to $P(-1, 2)$.For the line $x+2y+2=0$,the new coordinates are $x = X-1$ and $y = Y+2$.Substituting these into the line equation:$(X-1) + 2(Y+2) + 2 = 0$$X-1 + 2Y+4 + 2 = 0$$X+2Y+5 = 0$.

When the origin is shifted to the point $P$ by translation of axes,the equation $2x^2+y^2-4x+4y=0$ is transformed to $2x^2+y^2-8x+8y+18=0$. Then the transformed equation of the straight line $x+2y+2=0$ if the origin is shifted to the same point $P$ is

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