Question

determine the equation of the circle graphed below.

Explanation:

Step1: Recall the standard equation of a circle

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.

Step2: Identify the center of the circle

From the graph, the center of the circle is \((h,k)=(-5,2)\).

Step3: Calculate the radius

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) to find the radius. Let \((x_1,y_1)=(-5,2)\) (center) and \((x_2,y_2)=(-2,6)\) (a point on the circle). Then \(r=\sqrt{(-2+5)^2+(6 - 2)^2}=\sqrt{3^2+4^2}=\sqrt{9 + 16}=\sqrt{25}=5\).

Step4: Substitute \(h,k,r\) into the standard equation

Substitute \(h=-5,k = 2,r = 5\) into \((x - h)^2+(y - k)^2=r^2\). We get \((x+5)^2+(y - 2)^2=25\).

Answer:

\((x + 5)^2+(y - 2)^2=25\)