determine the equation of the circle graphed below. answer attempt 1 out of 2

Explanation:

Step1: Recall circle - equation formula

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 $(h,k)=(-5,6)$.

Step3: Calculate the radius

The radius $r$ is the distance between the center $(-5,6)$ and a point on the circle, say $(-2,8)$. Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, we have $r=\sqrt{(-2+5)^2+(8 - 6)^2}=\sqrt{3^2+2^2}=\sqrt{9 + 4}=\sqrt{13}$.

Step4: Substitute values into the circle - equation formula

Substitute $h=-5$, $k = 6$, and $r=\sqrt{13}$ into $(x - h)^2+(y - k)^2=r^2$. We get $(x + 5)^2+(y - 6)^2=13$.

Answer:

$(x + 5)^2+(y - 6)^2=13$