Question

the equation of a circle is $(x + 4)^{2}+(y - 5)^{2}=121$.
what is the center and radius of the circle?
center: $(-4,5)$; radius: 121
center: $(4,-5)$; radius: 11
center: $(-4,5)$; radius: 11
center: $(-4,-5)$; radius: 121

Explanation:

Step1: Recall the standard form of a circle's equation

The standard form of a circle's equation is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center and \(r\) is the radius.

Step2: Rewrite the given equation in the standard form

The given equation is \((x + 4)^2+(y - 5)^2=121\). We can rewrite \((x + 4)^2\) as \((x-(-4))^2\) and \(121\) as \(11^2\). So the equation becomes \((x-(-4))^2+(y - 5)^2=11^2\).

Step3: Identify the center and radius

Comparing with the standard form \((x - h)^2+(y - k)^2=r^2\), we have \(h=-4\), \(k = 5\), and \(r = 11\). So the center is \((-4,5)\) and the radius is \(11\).

Answer:

center: \((-4,5)\); radius: \(11\) (corresponds to the third option in the multiple - choice list)