The transformed equation of 2x^2 + 3y^2 - z^2 | vedclass

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Question

(A) Given the original equation: $2x^2 + 3y^2 - z^2 - 8x + 18y + 2z + 9 = 0$. \ The axes are translated to the point $(h, k, l) = (2, -3, 1)$. \ The

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$2x^2 + 3y^2 - z^2 = 25$

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(A) Given the original equation: $2x^2 + 3y^2 - z^2 - 8x + 18y + 2z + 9 = 0$. \ The axes are translated to the point $(h, k, l) = (2, -3, 1)$. \ The transformation equations are $x = X + 2$,$y = Y - 3$,and $z = Z + 1$. \ Substituting these into the original equation: \ $2(X + 2)^2 + 3(Y - 3)^2 - (Z + 1)^2 - 8(X + 2) + 18(Y - 3) + 2(Z + 1) + 9 = 0$ \ Expanding the terms: \ $2(X^2 + 4X + 4) + 3(Y^2 - 6Y + 9) - (Z^2 + 2Z + 1) - 8X - 16 + 18Y - 54 + 2Z + 2 + 9 = 0$ \ $2X^2 + 8X + 8 + 3Y^2 - 18Y + 27 - Z^2 - 2Z - 1 - 8X - 16 + 18Y - 54 + 2Z + 2 + 9 = 0$ \ Combining like terms: \ $2X^2 + 3Y^2 - Z^2 + (8X - 8X) + (-18Y + 18Y) + (-2Z + 2Z) + (8 + 27 - 1 - 16 - 54 + 2 + 9) = 0$ \ $2X^2 + 3Y^2 - Z^2 - 25 = 0$ \ Therefore,the transformed equation is $2X^2 + 3Y^2 - Z^2 = 25$.

The transformed equation of $2x^2 + 3y^2 - z^2 - 8x + 18y + 2z + 9 = 0$ when the axes are translated to the point $(2, -3, 1)$ is

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