Question

determine each of the following for the circle whose equation is ( x^{2}+y^{2}+10y - 11 = 0 ). make sure your answer is fully simplified.
the equation of the circle in standard (center, radius) form is:
the center of the circle is:
the radius of the circle is:
graph the circle:

Explanation:

Step1: Complete the square for \(y\) 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.
Given \(x^{2}+y^{2}+10y - 11=0\).
Group \(y\) terms: \(x^{2}+(y^{2}+10y)=11\).
For \(y^{2}+10y\), using the formula \((m + n)^2=m^{2}+2mn + n^{2}\), here \(m = y\), \(2mn=10y\Rightarrow n = 5\).
Add \(n^{2}=25\) to both sides: \(x^{2}+(y^{2}+10y + 25)=11 + 25\).

Step2: Write in standard form

\(x^{2}+(y + 5)^{2}=36\).

Answer:

• The equation of the circle in standard (center, radius) form is: \(x^{2}+(y + 5)^{2}=36\)
• The center of the circle is: \((0,-5)\)
• The radius of the circle is: \(6\)