Question

find the center and radius of the circle having the equation: x² - 10x + y² - 2y + 15 = 0

Explanation:

Step1: Complete the square for \(x\) terms

The general equation of a circle is \((x - a)^2+(y - b)^2=r^2\), where \((a,b)\) is the center and \(r\) is the radius.
For the given equation \(x^{2}-10x + y^{2}-2y+15 = 0\), complete the square for \(x\) terms:
\(x^{2}-10x=(x - 5)^2-25\) (using \((m - n)^2=m^{2}-2mn + n^{2}\), here \(m = x\), \(n = 5\))

Step2: Complete the square for \(y\) terms

Complete the square for \(y\) terms:
\(y^{2}-2y=(y - 1)^2-1\) (using \((m - n)^2=m^{2}-2mn + n^{2}\), here \(m = y\), \(n = 1\))

Step3: Rewrite the equation

Substitute the completed - square expressions into the original equation:
\((x - 5)^2-25+(y - 1)^2-1 + 15=0\)
\((x - 5)^2+(y - 1)^2-11 = 0\)
\((x - 5)^2+(y - 1)^2=11\)

Answer:

The center of the circle is \((5,1)\) and the radius is \(\sqrt{11}\)