To which point should the origin be shifted,w | vedclass

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(A) Let the new origin be $(h, k)$. The transformation equations are $x = X + h$ and $y = Y + k$.Substituting these into the given equation: $(X + h)^

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$(2, -3)$

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(A) Let the new origin be $(h, k)$. The transformation equations are $x = X + h$ and $y = Y + k$.Substituting these into the given equation: $(X + h)^2 + (Y + k)^2 - 4(X + h) + 6(Y + k) - 7 = 0$.Expanding this: $X^2 + 2hX + h^2 + Y^2 + 2kY + k^2 - 4X - 4h + 6Y + 6k - 7 = 0$.Grouping the terms: $X^2 + Y^2 + X(2h - 4) + Y(2k + 6) + (h^2 + k^2 - 4h + 6k - 7) = 0$.For the first-degree terms to vanish,the coefficients of $X$ and $Y$ must be zero:$2h - 4 = 0 \implies h = 2$.$2k + 6 = 0 \implies k = -3$.Thus,the origin should be shifted to $(2, -3)$.

To which point should the origin be shifted,without changing the direction of the axes,so that the equation $x^2 + y^2 - 4x + 6y - 7 = 0$ transforms into an equation that contains no first-degree terms?

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