Q 10. - 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 $\sqrt{2}$ units.

Standard equation of a circle is $(x-1{)}^2+(y-2{)}^2=2$.

Step 1. Given information

Figure is as follows:

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

Diameter is the distance between the points $(2,3)$ and $\left(0,1\right)$

Diameter $=\sqrt{(2-0{)}^2+(3-1{)}^2}$

                $\begin{array}{rcl}& =& \sqrt{4+4}\\ & =& 2\sqrt{2}\end{array}$

Center is the midpoint of the points $\left(2,3\right)$ and $\left(0,1\right)$

$\begin{array}{rcl}\text{ Center }& =& \left(\frac{2+0}{2},\frac{3+1}{2}\right)\\ & =& (1,2)\end{array}$

Step 3. Find the standard form of an equation of a circle.

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 $\sqrt{2}$ is

$(x-1{)}^2+(y-2{)}^2=(\sqrt{2}{)}^2$

$(x-1{)}^2+(y-2{)}^2=2$