Chapter 13.1, Problem 20PSB

To determine

To find: The slope of the tangent to the circle at $\left(7,8\right)$ .

Answer to Problem 20PSB

The slope of the radius line is $-1$ .

Explanation of Solution

Given:

It is given that the equation is $y-y_1=m\left(x-x_1\right)$ , containing point $\left(5,2\right)$ , and slope is 6.

Calculation:

Find the slope of the radius line radius with points $\left(2,3\right)$ and $\left(7,8\right)$ .

Substitute $2$ for $x_1$ , $7$ for $x_2$ , $8$ for $y_2$ and $3$ for $y_1$ in the equation $m=\frac{y_2-y_1}{x_2-x_1}$ .

  $\begin{array}{c}{m}_r=\frac{8-3}{7-2}\\ =\frac{5}{5}\\ =1\end{array}$

The tangent line is perpendicular to the radius line. So $m_t=-\frac{1}{m_r}$ .

  $\begin{array}{c}{m}_t=-\frac{1}{1}\\ =-1\end{array}$

Therefore, the slope of the radius line is $-1$ .