College Algebra, Chapter 2, Review Exercises, Section Review Exercises, Problem 64

Find equations for the circle and the line in the figure below



Recall that the general equation of the circle with center $(h,k)$ and radius $r$ is...
$(x - h)^2 + ( y - k)^2 = r^2$

By observation, the circle has center at $(5,5)$ and has radius $5$, thus the equation of the circle is...

$\begin{equation} \begin{aligned} (x-5)^2 + (y - 5) &= 5^2\\ \\ (x-5)^2 + (y-5)^2 &= 25 \end{aligned} \end{equation}$


Since the line passes through $(5,5)$ and $(8,1)$, by using two point form, we will be able to determine the equation of the line. So,

$\begin{equation} \begin{aligned} y - y_1 &= \left( \frac{y_2 - y_1}{x_2 - x_1} \right) (x - x_1)\\ \\ y -5 &= \left( \frac{1-5}{8-5} \right) (x - 5)\\ \\ y -5 &= \left( \frac{-4}{3} \right) (x - 5)\\ \\ y -5 &= -\frac{4}{3} x + \frac{20}{3}\\ \\ y &= -\frac{4}{3} x + \frac{20}{3}+5\\ \\ y &= -\frac{4}{3} x + \frac{35}{3} \end{aligned} \end{equation}$