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(C) Let the origin be shifted to $P(h, k)$. The transformation equations are $x = x' + h$ and $y = y' + k$.Substituting these into the first equation

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$2x^2+3xy-5y^2=0$

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(C) Let the origin be shifted to $P(h, k)$. The transformation equations are $x = x' + h$ and $y = y' + k$.Substituting these into the first equation $3x^2 + y^2 - 6x + 4y + 4 = 0$:$3(x' + h)^2 + (y' + k)^2 - 6(x' + h) + 4(y' + k) + 4 = 0$$3(x'^2 + 2hx' + h^2) + (y'^2 + 2ky' + k^2) - 6x' - 6h + 4y' + 4k + 4 = 0$$3x'^2 + y'^2 + (6h - 6)x' + (2k + 4)y' + (3h^2 + k^2 - 6h + 4k + 4) = 0$.To remove the first-degree terms,set the coefficients of $x'$ and $y'$ to zero:$6h - 6 = 0 \Rightarrow h = 1$$2k + 4 = 0 \Rightarrow k = -2$.So,the origin is shifted to $P(1, -2)$.Now,substitute $x = x' + 1$ and $y = y' - 2$ into the second equation $2x^2 + 3xy - 5y^2 + 2x - 23y - 24 = 0$:$2(x' + 1)^2 + 3(x' + 1)(y' - 2) - 5(y' - 2)^2 + 2(x' + 1) - 23(y' - 2) - 24 = 0$$2(x'^2 + 2x' + 1) + 3(x'y' - 2x' + y' - 2) - 5(y'^2 - 4y' + 4) + 2x' + 2 - 23y' + 46 - 24 = 0$$2x'^2 + 4x' + 2 + 3x'y' - 6x' + 3y' - 6 - 5y'^2 + 20y' - 20 + 2x' - 23y' + 46 - 24 = 0$Grouping the terms:$2x'^2 + 3x'y' - 5y'^2 + (4 - 6 + 2)x' + (3 + 20 - 23)y' + (2 - 6 - 20 + 46 - 24) = 0$$2x'^2 + 3x'y' - 5y'^2 + 0x' + 0y' + 0 = 0$.Thus,the transformed equation is $2x^2 + 3xy - 5y^2 = 0$.

Let $P$ be the point to which the origin is shifted by the translation of axes so as to remove the first-degree terms from the equation $3x^2+y^2-6x+4y+4=0$. If the origin is shifted to $P$ by the translation of axes,then the transformed equation of $2x^2+3xy-5y^2+2x-23y-24=0$ is

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