In order to eliminate the first degree terms | vedclass

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Question

(C) The given equation is $2x^2 + 4xy + 5y^2 - 4x - 22y + 7 = 0$. Comparing this with the general second-degree equation $ax^2 + 2hxy + by^2 + 2gx + 2

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$(-2, 3)$

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(C) The given equation is $2x^2 + 4xy + 5y^2 - 4x - 22y + 7 = 0$. Comparing this with the general second-degree equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$,we get $a = 2, h = 2, b = 5, g = -2, f = -11, c = 7$. To eliminate the first-degree terms,the origin $(0, 0)$ must be shifted to the point $(h_0, k_0)$ given by the intersection of the partial derivatives $\frac{\partial f}{\partial x} = 0$ and $\frac{\partial f}{\partial y} = 0$. These are given by the formulas: $h_0 = \frac{hf - bg}{ab - h^2} = \frac{(2)(-11) - (5)(-2)}{(2)(5) - (2)^2} = \frac{-22 + 10}{10 - 4} = \frac{-12}{6} = -2$ $k_0 = \frac{gh - af}{ab - h^2} = \frac{(-2)(2) - (2)(-11)}{(2)(5) - (2)^2} = \frac{-4 + 22}{10 - 4} = \frac{18}{6} = 3$ Thus,the origin should be shifted to the point $(-2, 3)$.

In order to eliminate the first degree terms from the equation $2x^2 + 4xy + 5y^2 - 4x - 22y + 7 = 0$,the point to which the origin is to be shifted is:

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