Q. 37 - Geometry

Question

 In Exercises 35-38 find an equation of the circle described and sketch the graph.

 The circle has diameter RS where R is (-3, 2) and S is (3, 2).

Step-by-Step Solution

Answer

The equation of the circle is$x^2+{\left(y-2\right)}^2=9$ .

The graph is:

Step-1 – Given

Given that the circle has diameter RS where $R\text{ = }\left(-3,2\right)\text{ and S = }\left(3,2\right)$

Step-2 – To determine

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

Step-3 – Calculation

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

$\begin{array}{l}RS=\sqrt{{\left(3-\left(-3\right)\right)}^2+{\left(2-2\right)}^2}\\ RS=\sqrt{{\left(3+3\right)}^2+{\left(0\right)}^2}\\ RS=\sqrt{{\left(6\right)}^2}\\ RS=6\end{array}$

So, diameter = 6.

It means, radius = $\frac{6}{2}=3$

Using the midpoint formula, we find the center:

$\begin{array}{l}c=\frac{R+S}{2}\left[c=\text{center}\right]\\ c=\frac{-3+3}{2},\frac{2+2}{2}\\ c=\frac{0}{2},\frac{4}{2}\\ c=\left(0,2\right)\end{array}$

Here, center = (a, b) = (0, 2) and radius = r = 3.

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

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

${\left(x-0\right)}^2+{\left(y-2\right)}^2=3^2$

$x^2+{\left(y-2\right)}^2=9$

So, the equation of the circle is $x^2+{\left(y-2\right)}^2=9$

Step-4 – Graph

 We will sketch 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 $x^2+{\left(y-2\right)}^2=9$ .

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 (0, 2) and the radius is 3.