Chapter 13.1, Problem 21PSC

(a)

To determine

To find: The slope of $\overleftrightarrow{\text{PT}}$ .

Answer to Problem 21PSC

The slope of the radius line is $-\frac{5}{8}$ .

Explanation of Solution

Given:

The given figure is as follows.

  

Calculation:

Find the slope of the radius line radius with points $\left(3,1\right)$ and $\left(8,9\right)$ .

Substitute $3$ for $x_1$ , $8$ for $x_2$ , $9$ for $y_2$ and $1$ for $y_1$ in the equation $m=\frac{y_2-y_1}{x_2-x_1}$ .

  $\begin{array}{c}{m}_r=\frac{9-1}{8-3}\\ =\frac{8}{5}\end{array}$

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

  $\begin{array}{c}{m}_t=-\frac{1}{\frac{8}{5}}\\ =-\frac{5}{8}\end{array}$

Therefore, the slope of the radius line is $-\frac{5}{8}$ .

(b)

To determine

To verify: That $y-9=-\frac{5}{8}\left(x-8\right)$ is an equation that represents $\overleftrightarrow{\text{PT}}$ .

Answer to Problem 21PSC

It is verified that the equation $y-9=-\frac{5}{8}\left(x-8\right)$ is an equation that represents $\overleftrightarrow{\text{PT}}$ .

Explanation of Solution

Given:

The given figure is as follows.

  

Calculation:

Substitute $8$ for $x$ , and $9$ for $y$ in the equation $y-9=-\frac{5}{8}\left(x-8\right)$ to verify that $y-9=-\frac{5}{8}\left(x-8\right)$ is an equation that represents $\overleftrightarrow{\text{PT}}$ .

  $\begin{array}{c}9-9=-\frac{5}{8}\left(8-8\right)\\ 0=0\end{array}$

Therefore, it is verified that the equation $y-9=-\frac{5}{8}\left(x-8\right)$ is an equation that represents $\overleftrightarrow{\text{PT}}$ .

(c)

To determine

To verify: That the equation $y=-\frac{5}{8}x+14$ is an equation that represents $\overleftrightarrow{\text{PT}}$ .

Answer to Problem 21PSC

It is verified that the equation $y=-\frac{5}{8}x+14$ is an equation that represents $\overleftrightarrow{\text{PT}}$ .

Explanation of Solution

Given:

The given figure is as follows.

  

Calculation:

Substitute $8$ for $x$ , and $9$ for $y$ in the equation $y=-\frac{5}{8}x+14$ to verify that $y=-\frac{5}{8}x+14$ is an equation that represents $\overleftrightarrow{\text{PT}}$ .

  $\begin{array}{c}9=-\frac{5}{8}\left(8\right)+14\\ 9=-5+14\\ 9=9\end{array}$

Therefore, it is verified that the equation $y=-\frac{5}{8}x+14$ is an equation that represents $\overleftrightarrow{\text{PT}}$ .