Equations of Circles Solutions

• Methods for finding the radius of a circle:

  • On the coordinate plane,    estimate    its length. 

  • Calculate using the    distance formula   , if you have the    center    of a circle and a    point    on the circle.
• 
Methods for finding the center of a circle: 

  • On the coordinate plane, find the    midpoint    of either the horizontal or vertical diameter.
  • 
Calculate using the    midpoint formula    if you have the endpoints of the    diameter   .

Example 3

Determine the equation of the circle with center $\mathit{C}\mathbf{ }\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{8}\mathbf{,}\mathbf{ }\mathbf{–}\mathbf{1}\mathbf{.}\mathbf{6}\mathbf{)}$ and $\mathit{P}\mathbf{ }\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{8}\mathbf{,}\mathbf{ }\mathbf{–}\mathbf{0}\mathbf{.}\mathbf{62}\mathbf{)}$ on the circle. Round the length of the radius to the nearest tenth.

$$\begin{gathered} r=\sqrt{{\left(1.8–0.8\right)}^2+{\left(–0.62–\left(–1.6\right)\right)}^2} \\ r=\sqrt{{\left(1\right)}^2+{\left(0.98\right)}^2} \\ r=1.4 \\ {\left(x–0.8\right)}^2+{\left(y–\left(–1.6\right)\right)}^2=1.4^2 \\ {\left(x–0.8\right)}^2+{\left(y+1.6\right)}^2=1.96 \end{gathered}$$

Example 4

Determine the equation of the circle with the endpoints of the diameter being $\mathbf{(}\mathbf{–}\mathbf{5}\mathbf{,}\mathbf{ }\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{)}$ and $\mathbf{(}\mathbf{–}\mathbf{2}\mathbf{,}\mathbf{ }\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{)}$. Name the domain and range of the circle in interval notation.