Chapter 13.3, Problem 20WE

To determine

To find the slope of the line that is tangent to the circle at P.

Answer to Problem 20WE

Slope of the tangent is $\frac{-4}{3}$ .

Explanation of Solution

Given information: The points $P\left(6,7\right)$ lies on the circle ${\left(x+2\right)}^2+{\left(y-1\right)}^2=100$ .

Calculation: The required solutions is obtained as,

  

C is the center of the circle, $C=\left(-2,1\right)$ .

Slope,

  $m_{CP}=\frac{7-1}{6-\left(-2\right)}=\frac{6}{8}=\frac{3}{4}$

CP is perpendicular to PQ.

  $\begin{array}{l}{m}_{CP}\times m_{PQ}=-1\\ \frac{3}{4}\times m_{PQ}=-1\\ m_{PQ}=\frac{-4}{3}\end{array}$

Hence, slope of the tangent is $\frac{-4}{3}$ .