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x^(2)-2x+y^(2)-4y-4=0

$x^{2}-2 x+y^{2}-4 y-4=0$

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Find derivatives of using multiple formulae

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Q.

x^{2}-2 x+y^{2}-4 y-4=0

Complete the square for $x$ : To find the center and radius of the circle, we need to complete the square for both $x$ and $y$ terms in the equation $x^{2}-2x+y^{2}-4y-4=0$ .
Group $x$ and $y$ terms: First, we group the $x$ terms and the $y$ terms together: $(x^2 - 2x) + (y^2 - 4y) = 4$ .
Add values to complete square: Next, we complete the square for the $x$ terms. We take half of the coefficient of $x$ , which is $-\frac{2}{2} = -1$ , square it to get $1$ , and add it to both sides of the equation: $(x^2 - 2x + 1) + (y^2 - 4y) = 4 + 1$ .
Standard form of circle equation: Now, we complete the square for the y terms. We take half of the coefficient of y, which is $-4/2 = -2$ , square it to get $4$ , and add it to both sides of the equation: $(x^2 - 2x + 1) + (y^2 - 4y + 4) = 4 + 1 + 4$ .
Identify center and radius: After adding the necessary values to complete the square, we have: $(x - 1)^2 + (y - 2)^2 = 9$ .
Identify center and radius: After adding the necessary values to complete the square, we have: $(x - 1)^2 + (y - 2)^2 = 9$ .The equation $(x - 1)^2 + (y - 2)^2 = 9$ is now in the standard form of a circle's equation, $(x - h)^2 + (y - k)^2 = r^2$ , where $(h, k)$ is the center and $r$ is the radius.
Identify center and radius: After adding the necessary values to complete the square, we have: $(x - 1)^2 + (y - 2)^2 = 9$ .The equation $(x - 1)^2 + (y - 2)^2 = 9$ is now in the standard form of a circle's equation, $(x - h)^2 + (y - k)^2 = r^2$ , where $(h, k)$ is the center and $r$ is the radius.From the equation $(x - 1)^2 + (y - 2)^2 = 9$ , we can see that the center $(h, k)$ of the circle is $(1, 2)$ and the radius $r$ is the square root of $9$ , which is $(x - 1)^2 + (y - 2)^2 = 9$ $0$ .

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