Q 8. - 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 $\left(1,2\right)$.

Radius is equal to $2$ units.

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

Step 1. Given information

Figure of a circle is 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 $(1,2)$ and $\left(1,0\right)$.

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

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

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

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 $\left(1,2\right)$ and radius $2$ is

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