Chapter 13.3, Problem 19WE

To determine

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

Answer to Problem 19WE

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

Explanation of Solution

Given information: The points $N\left(3,-4\right)$ lies on the circle $x^2+y^2=25$ .

Calculation: The required solutions is obtained as,

  

Line PQ is tangent at P.

So, OP is perpendicular to PQ.

  $\begin{array}{l}{m}_{OP}\times m_{PQ}=-1\\ m_{OP}=\frac{-4-0}{3-0}=\frac{-4}{3}\\ m_{OP}\times m_{PQ}=-1\\ \frac{-4}{3}\times m_{PQ}=-1\\ m_{PQ}=\frac{3}{4}\end{array}$

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