Chapter 13.1, Problem 38WE

To determine

To find the equation of the circle and sketch the graph

Answer to Problem 38WE

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

Explanation of Solution

Given information:

The circle has center $\left(p,q\right)$ and is tangent to the x- axis

Calculation:

The equation of the circle can be obtained as

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

So, $\left(a,b\right)=\left(p,q\right)$

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

Since the tangent to the x- axis so, $y=0$

Therefore,

  $S\left(x,0\right)=S\left(p,0\right)$

First find the radius

Put the values of $a=p,b=q,x=p\text{ and }y=0$ on the standard form of the equation

  $\begin{array}{l}r=\sqrt{{\left(p-p\right)}^2+{\left(q-0\right)}^2}\\ r=\sqrt{{\left(0\right)}^2+{\left(q\right)}^2}\\ r=\sqrt{{\left(q\right)}^2}\\ r=q\end{array}$

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

  ${\left(x-p\right)}^2+{\left(y-q\right)}^2={\left(q\right)}^2$

Hence,

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

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