91Ó°ÊÓ

Given that the circle has center (p, q) and is tangent to the x-axis.

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

We first find the length of the radius:

Let, S(x, y) is the tangent point on the circle.

Since the tangent is the x-axis so on x-axis y = 0.

It means, S(x, 0) = S(p, 0).

Then we find the radius by finding the distance between (p, q) and (p, 0).

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

It means, radius = r = q.

Here, center = (a, b) = (p, q) and radius = r = q.

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

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

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

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

So, the equation of the circle is${\left(x−p\right)}^2+{\left(y−q\right)}^2=q^2$

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${\left(x−p\right)}^2+{\left(y−q\right)}^2=q^2$ . (used p = 4 and q = 3).

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 (p, q) and the radius is q.