Chapter 13.1, Problem 36WE

To determine

To find the equation of the circle and sketch the graph

Answer to Problem 36WE

The equation of the circle are ${\left(x+2\right)}^2+{\left(y+4\right)}^2=169$

Explanation of Solution

Given information:

The circle has center $\left(-2,-4\right)$ and passes through the point $\left(3,8\right)$

Calculation:

The equation of the circle can be obtained as

Let $P\left(x,y\right)$ represent the point on the circle

So, $P\left(x,y\right)=P\left(3,8\right)$

The standard form of the equation of a circle with center is

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

Where $\left(a,b\right)$ is the center and r is the radius

Put the values of $a=-2,b=-4,x=3\text{ and }y=8$ on the standard form the equation

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

Again put the values of $a=-2,b=-4\text{ and }r=13$ on the standard form of the equation

  $\begin{array}{l}{\left(x-\left(-2\right)\right)}^2+{\left(y-\left(-4\right)\right)}^2={\left(13\right)}^2\\ {\left(x+2\right)}^2+{\left(y+4\right)}^2=169\end{array}$

Hence,

The equation of the circle are ${\left(x+2\right)}^2+{\left(y+4\right)}^2=169$

Graph : Sketch the graph using graphing utility.

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

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

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

Step 4: Press GRAPH button to graph the function and adjust the windows according to the graph.

The graph is obtained as:

  

Interpretation :

From the above graph, it can be observed that the center of a circle is $\left(-2,-4\right)$ and radius is 13.