When the origin is shifted to the point (2, b | vedclass

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(D) Given that the origin is shifted to $(2, b)$,the transformation equations are $x = X + 2$ and $y = Y + b$. Since the point $(a, 4)$ changes to $(6

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$-C$

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(D) Given that the origin is shifted to $(2, b)$,the transformation equations are $x = X + 2$ and $y = Y + b$. Since the point $(a, 4)$ changes to $(6, 8)$,we have $a = 6 + 2 = 8$ and $4 = 8 + b$,which gives $b = -4$. Now,the origin is shifted to $(a, b) = (8, -4)$,so the transformation equations are $x = X + 8$ and $y = Y - 4$. Substituting these into the equation $x^2 + 4xy + y^2 = 0$: $(X + 8)^2 + 4(X + 8)(Y - 4) + (Y - 4)^2 = 0$ $(X^2 + 16X + 64) + 4(XY - 4X + 8Y - 32) + (Y^2 - 8Y + 16) = 0$ $X^2 + 16X + 64 + 4XY - 16X + 32Y - 128 + Y^2 - 8Y + 16 = 0$ $X^2 + 4XY + Y^2 + 24Y - 48 = 0$ Comparing this with $X^2 + 2HXY + Y^2 + 2GX + 2FY + C = 0$,we get $2H = 4 \Rightarrow H = 2$,$2G = 0 \Rightarrow G = 0$,$2F = 24 \Rightarrow F = 12$,and $C = -48$. Finally,$2H(G + F) = 2(2)(0 + 12) = 4(12) = 48$. Since $C = -48$,$48 = -C$.

When the origin is shifted to the point $(2, b)$ by translation of axes,the coordinates of the point $(a, 4)$ change to $(6, 8)$. When the origin is shifted to $(a, b)$ by translation of axes,if the transformed equation of $x^2+4xy+y^2=0$ is $X^2+2HXY+Y^2+2GX+2FY+C=0$,then $2H(G+F)=$

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