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Q. 2

Expert-verified
Found in: Page 556

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# Find an equation for the circle with center at the point $\left(-5,1\right)$ and radius 3. Graph this circle.

• The equation of the circle is $\begin{array}{rcl}{\left(x+5\right)}^{2}+{\left(y-1\right)}^{2}& =& 9\end{array}$.
• The graph is

See the step by step solution

## Step 1. Given Information

• The center of the circle is $\left(-5,1\right)$.

## Step 2. Find the Equation

• The general equation of a circle is ${\left(x-h\right)}^{2}+{\left(x-k\right)}^{2}={r}^{2}$.
• Substitute $h=-5,k=1$ into the general equation and simplify.

localid="1647407263863" $\begin{array}{rcl}\left(x-{\left(-5\right)\right)}^{2}+{\left(y-1\right)}^{2}& =& {3}^{2}\\ {\left(x+5\right)}^{2}+{\left(y-1\right)}^{2}& =& 9\end{array}$

## Step 3. Find the Intercepts

• Substitute 0 for x and find the y-intercepts.

$\begin{array}{rcl}{\left(0+5\right)}^{2}+{\left(y-1\right)}^{2}& =& 9\\ {\left(y-1\right)}^{2}& =& -16\end{array}$

• As the square is a negative value, there are no x-intercepts.
• Substitute 0 for y and find the x-intercepts.

$\begin{array}{rcl}{\left(x+5\right)}^{2}+{\left(0-1\right)}^{2}& =& 9{\left(x+5\right)}^{2}=8x+5=±2\sqrt{2}\\ x& =& -2\sqrt{2}-5,-2\sqrt{2}+5\\ & & \end{array}$

## Step 4. Plot the Graph

Use the center and the intercepts to plot the circle.