Chapter 16.1, Problem 8PS

To determine

To show: Thatthe graph of the line tangent to the given circle.

Explanation of Solution

Given information:

The given equation of the line is,

  $12x+5y=12$

The centre of the circle is at $\left(6,1\right)$ and passing through $\left(9,-3\right)$ .

Calculate the radius of the circle as,

  $\begin{array}{l}{r}^2={\left(x-h\right)}^2+{\left(y-k\right)}^2\\ r^2={\left(9-6\right)}^2+{\left(-3-1\right)}^2\\ r=5\text{unit}\end{array}$

The distance from the centre to the graph of the given equation of the line is calculated as,

  $\begin{array}{c}d=\frac{\left|a{x}_1+b{y}_1+c\right|}{\sqrt{a^2+b^2}}\\ =\frac{\left|12\left(6\right)+\left(5\right)\left(1\right)+\left(-12\right)\right|}{\sqrt{{\left(12\right)}^2+{\left(5\right)}^2}}\\ =\frac{65}{13}\\ =5\text{units}\end{array}$

The radius of the circle is $5\text{units}$ and the distance from the centre to the graph of the given equation of the line is $5\text{units}$ , so the graph of the line tangent to the given circle.

Thus, the graph of the line is tangent to the given circle.