Question

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

Short answer

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

Step-by-step solution

Step 1. Given Information

• The center of the circle is $(-5,1)$.
• The radius is 3.

Step 2. Find the Equation

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

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

Step 3. Find the Intercepts

• Substitute 0 for xand find the y-intercepts.

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

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

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

Step 4. Plot the Graph

Use the center and the intercepts to plot the circle.