Q. 70 - Precalculus Enhanced with Graphing Utilities

Question

In Problems $65-70$, graph each equation and find the point(s) of intersection, if any.

$y=\frac{4}{x+2}$ and the circle $x^2+4x+y^2-4=0$.

Step-by-Step Solution

Answer

The graph of the line $y=\frac{4}{x+2}$ and the circle $x^2+4x+y^2-4=0$ is,

From the graph, the intersecting points are, $\left(-4,-2\right),\left(0,2\right)$.

Step 1. Given information

$$\begin{gathered} y=\frac{4}{x+2} \\ {x}^2+4x+y^2-4=0 \end{gathered}$$

First solve the equation for $y$.

Consider the first equation,

$y=\frac{4}{x+2}$

Consider the second equation,

$$\begin{gathered} x^2+4x+y^2-4=0 \\ {y}^2=-x^2-4x+4 \\ y=\pm \sqrt{-x^2-4x+4} \end{gathered}$$

Step 2 Now sketch a graph by using the obtained equations.

In the graph, the two equations are intersecting at two points. They are, $\left(-4,-2\right),\left(0,2\right)$.