Chapter 8, Problem 6TP

Solution

To graph: The number of solutions of the system that the two hyperbola one with a vertical transverse axis and other with a horizontal transverse axis represents.

Given information:

A coordinated grid contains two hyperbolas centered at the origin, one with a vertical transverse axis and other with a horizontal transverse axis represent.

Calculation:

The transverse axis is the axis of the hyperbola which passes through its two vertices. Notice that if $(x,y)$ is a solution of the system, then $(\pm x,\pm y)$ are also solutions.

Hence, the system either has no solutions or it has four solutions (because a system of two equations with two unknowns can have at most $2$ . $2=4$ solutions).

The number of solutions could be $1,2or3$ only if $x=0$ or $y=0$ , however it is not possible that the hyperbolas centered at $(0,0)$ with axes on the coordinate axes intersect at points on the axes (these are only the four vertices of the hyperbolas).

The system can have no solutions, for example

  $\begin{array}{r}\hfill \frac{x^2}{9}-\frac{y^2}{4}=1\\ \hfill -\frac{x^2}{4}+\frac{y^2}{9}=1\end{array}$

The two hyperbolas don't intersect.

The system can also have four solutions, for example

  $\begin{array}{r}\hfill \frac{x^2}{9}-\frac{y^2}{4}=1\\ \hfill -x^2+\frac{y^2}{0.25}=1\end{array}$

The two hyperbolas intersect at four points $A,B,C,$ and $D$

Interpretation:

The two hyperbolas don't intersect.

The two hyperbolas intersect at four points $A,B,C,$ and $D$

To graph The two hyperbolas don't intersect

Given information : The two hyperbolas don't intersect.

Graph :

Interpretation : The two hyperbolas don't intersect.

To graph The two hyperbolas intersect at four points $A,B,C,$ and $D$

Given information : The two hyperbolas intersect at four points $A,B,C,$ and $D$

Graph :

Interpretation : The two hyperbolas intersect at four points $A,B,C,$ and $D$