91Ó°ÊÓ

Given that the circle has center = (-2, -4) and passes through the point (3, 8).

We have to find the equation of the circle and sketch the graph.

We first find the length of the radius by finding the distance between (-2, -4) and (3, 8).

$\begin{array}{l}r=\sqrt{{\left(3−\left(−2\right)\right)}^2+{\left(8−\left(−4\right)\right)}^2}\\ r=\sqrt{{\left(3+2\right)}^2+{\left(8+4\right)}^2}\\ r=\sqrt{{\left(5\right)}^2+{\left(12\right)}^2}\\ r=\sqrt{25+144}\\ r=\sqrt{169}\\ r=13\end{array}$

Here, center = (a, b) = (-2, -4) and radius = r = 13.

We plug them in the standard form of the equation of a circle:

${\left(x−a\right)}^2+{\left(y−b\right)}^2=r^2$

${\left(x+2\right)}^2+{\left(y+4\right)}^2={13}^2$

${\left(x+2\right)}^2+{\left(y+4\right)}^2=169$

So, the equation of the circle is ${\left(x+2\right)}^2+{\left(y+4\right)}^2=169$.

We willsketch the graph using a graphing utility.

Step 1: Press WINDOW button in order to access the Window editor.

Step 2: Press$\boxed{\text{Y=}}$ button.

Step 3: Enter the expression ${\left(x+2\right)}^2+{\left(y+4\right)}^2=169$.

Step 4: Press GRAPH button to graph the function and then adjust the window.

The obtained graph is:

From the graph, we see that the center is (-2, -4) and the radius is 13.