Chapter 3.5, Problem 56E

To determine

To Explain: How you find the other point of intersection without performing any calculation?

Answer to Problem 56E

  $(-x,-y)$

Explanation of Solution

Given:Suppose the graph of the line that passes through the origin intersect the circle with its center at the origin.

Let the known point be $Q=(-x,-y)$

  

The equations formed be

  $\begin{array}{l}{x}^2+y^2=r^2\\ y=ax\end{array}$

Solving the equations.

  $\begin{array}{l}$ x^2+{(ax)}^2=r^2$x^2+a^2{x}^2=r^2\\ x^2(1+a^2)=r^2\\ x^2=\frac{r^2}{1+a^2}\\ now\\ x=\pm \frac{r}{\sqrt{1+a^2}}\\ x_1=-\frac{r}{\sqrt{1+a^2}}\\ y_1=\frac{-ar}{\sqrt{1+a^2}}\\ x_2=\frac{r}{\sqrt{1+a^2}}\\ y_2=\frac{ar}{\sqrt{1+a^2}}\end{array}$

From here we noticed that two points are symmetrical across the origin.