Q 7. - Precalculus Enhanced with Graphing Utilities

Question

Find the center and radius of the circle. Write the standard form of the equation.

Step-by-Step Solution

Answer

Center is $\left(2,1\right)$.

Radius is equal to $2$ units.

Standard form of the equation is $(x-2{)}^2+(y-1{)}^2=4$.

Step 1. Given information

Figure of a circle is given as follows:

Step 2. Find the center and radius of a circle.

Radius is the distance between the center and any point on the circle

In this case, radius is the distance between the points $(0,1)$ and $(2,1)$

Radius$=\sqrt{(0-2{)}^2+(1-1{)}^2}$

           $$\begin{gathered} =\sqrt{4+0} \\ =\sqrt{4} \\ =2 \end{gathered}$$

From the given graph, we can see that the center is $\left(2,1\right)$.

Step 3. Find the standard form of the equation.

Equation of a circle with center $\left(h,k\right)$ and radius $r$ is $(x-h{)}^2+(y-k{)}^2=r^2$

Therefore, equation of the circle with center $\left(2,1\right)$ and radius $2$ is

$\begin{array}{rcl}(x-2{)}^2+(y-1{)}^2& =& 2^2\\ (x-2{)}^2+(y-1{)}^2& =& 4\end{array}$