Q 34. - Precalculus Enhanced with Graphing Utilities

Question

Show that the points $A\left(1,5\right) ,B\left(-3,5\right)$ and

$C\left(-3,5\right)$ lie on a circle with center $\left(-1,2\right)$. What is the radius of the circle.

Step-by-Step Solution

Answer

Since the distance of all points from center is same so they are all lie on same circle and radius is $\sqrt{13}$.

Step 1. Given information .

Consider the given points $A\left(1,5\right) ,B\left(2,4\right)$ and $C\left(-3,5\right)$ and center $O\left(-1,2\right)$.

Step 2. Using distance formula to check the distance of points from center. d = x 2 - x 1 2 + y 2 - y 1 2

The formula used to find distance.

$$\begin{gathered} d\left(OA\right)=\sqrt{{\left(1+1\right)}^2+{\left(5-2\right)}^2} \\ d\left(OA\right)=\sqrt{13} \\ d\left(OB\right)=\sqrt{{\left(2+1\right)}^2+{\left(4-2\right)}^2} \\ d\left(OB\right)=\sqrt{13} \end{gathered}$$

Further simplify.

$$\begin{gathered} d\left(OC\right)=\sqrt{{\left(-3+1\right)}^2+{\left(2-5\right)}^2} \\ d\left(OC\right)=\sqrt{13} \end{gathered}$$