Chapter 13.1, Problem 35WE

To determine

To find the equation of the circle and sketch the graph

Answer to Problem 35WE

The equation of the circle are ${\left(x-0\right)}^2+{\left(y-6\right)}^2={10}^2$

Explanation of Solution

Given information:

The circle has center $\left(0,6\right)$ and passes through the point $\left(6,14\right)$

Calculation:

The equation of the circle can be obtained as

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

So, $P\left(x,y\right)=P\left(6,14\right)$

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 $a=0,b=6,x=6\text{ and }y=14$ on the standard form the equation

  $\begin{array}{l}r=\sqrt{{\left(6-0\right)}^2+{\left(14-6\right)}^2}\\ r=\sqrt{{\left(6\right)}^2+{\left(8\right)}^2}\\ r=\sqrt{36+64}\\ r=\sqrt{100}\\ r=10\end{array}$

Again put the values of $a=0,b=6\text{ and }r=10$ on the standard form of the equation

  ${\left(x-0\right)}^2+{\left(y-6\right)}^2={10}^2$

Hence,

The equation of the circle are ${\left(x-0\right)}^2+{\left(y-6\right)}^2={10}^2$

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 ${\left(x-0\right)}^2+{\left(y-6\right)}^2={10}^2$ 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 $\left(0,6\right)$ and radius is 10.