Question

give the center and radius of the circle represented by the equation x² + y² - 10x + 2y = - 22. the center of the circle is (type an ordered pair, using integers or fractions). the radius of the circle is (type an integer or a simplified fraction).

Explanation:

Step1: Complete the square for x - terms

The original equation is \(x^{2}+y^{2}-10x + 2y=-22\). For the \(x\) - terms \(x^{2}-10x\), we have \(x^{2}-10x=(x - 5)^{2}-25\) (using the formula \((a - b)^2=a^{2}-2ab + b^{2}\), here \(a = x\) and \(b = 5\)).

Step2: Complete the square for y - terms

For the \(y\) - terms \(y^{2}+2y\), we have \(y^{2}+2y=(y + 1)^{2}-1\) (using the formula \((a + b)^2=a^{2}+2ab + b^{2}\), here \(a = y\) and \(b = 1\)).

Step3: Rewrite the equation of the circle

Substitute the completed - square expressions into the original equation: \((x - 5)^{2}-25+(y + 1)^{2}-1=-22\).
Rearrange the equation: \((x - 5)^{2}+(y + 1)^{2}=-22 + 25+1\).

Step4: Simplify the right - hand side

Calculate \(-22 + 25+1=4\). So the equation of the circle is \((x - 5)^{2}+(y + 1)^{2}=4\).
The standard form of the equation of a circle is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center of the circle and \(r\) is the radius.

Answer:

The center of the circle is \((5,-1)\)
The radius of the circle is \(2\)