Chapter 13.6, Problem 13PSB

To determine

To find: The distance between the point of intersection of the graph of the given equation.

Answer to Problem 13PSB

The distance between the intersection points is $5\sqrt{2}$ .

Explanation of Solution

Given:

The given system of the equation is,

  $\begin{array}{l}{x}^2+y^2=17\\ x+y=3\end{array}$

Calculation:

Consider the given system of the equation is,

  $\begin{array}{l}{x}^2+y^2=17\\ x+y=3\end{array}$

Consider the second equation as,

  $\begin{array}{l}x+y=3\\ y=3-x\end{array}$

Substitute $y=3-x$ in the first equation as,

  $\begin{array}{l}{x}^2+{\left(3-x\right)}^2=17\\ 2x^2-6x+9=17\\ x^2-3x=4\\ x=4,-1\end{array}$

Then, the value of $y$ is obtained as,

  $\begin{array}{l}4+y=3\\ y=-1\end{array}$ or $\begin{array}{l}-1+y=3\\ y=4\end{array}$

Thus, the intersection points are $\left(4,-1\right),\left(-1,4\right)$ .

The distance between the intersection points is obtained as,

  $\begin{array}{c}d=\sqrt{{\left(4-\left(-1\right)\right)}^2+{\left(-1-4\right)}^2}\\ =\sqrt{25+25}\\ =5\sqrt{2}\end{array}$