Chapter 11.5, Problem 15E

(a)

To determine

To identify : whether the graph is parabola, an ellipse or a hyperbola

Answer to Problem 15E

The graph is hyperbola

Explanation of Solution

Given information :

The equation is $xy=8$

Calculation :

Compare $xy-8=0$ with standard equation of the graph $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$ ,

Since, $A=0,B=1,C=0$ so the discriminant is:

  $B^2-4AC=1^2-4\left(0\right)\left(0\right)=1>0$

So, the graph is hyperbola

(b)

To determine

To eliminate the $xy$ term using rotation axis.

Answer to Problem 15E

The equation is $x^2-y^2=16$

Explanation of Solution

Given information :

The equation is $xy=8$

Calculation :

To eliminate the $xy$ term

Let,

  $\begin{array}{l}\cot 2\varphi =\frac{a-c}{b}=0\\ 2\varphi =\frac{\pi }{2}\Rightarrow \varphi =\frac{\pi }{4}\\ X=x\cos \varphi -y\sin \varphi \\ X=x\cos \left(\frac{\pi }{4}\right)-y\sin \left(\frac{\pi }{4}\right)\Rightarrow \frac{x-y}{\sqrt{2}}\\ Y=x\sin \varphi +\cos \varphi \\ Y=x\sin \left(\frac{\pi }{4}\right)+\cos \left(\frac{\pi }{4}\right)\Rightarrow \frac{x+y}{\sqrt{2}}\\ xy=8\\ \Rightarrow \left(\frac{x-y}{\sqrt{2}}\right)\left(\frac{x+y}{\sqrt{2}}\right)=8\\ \Rightarrow \left(\frac{x^2-y^2}{2}\right)=8\\ \Rightarrow x^2-y^2=16\end{array}$

(c)

To determine

To sketch : the graph the equation $xy=8$

Explanation of Solution

Given information : The equation is $xy=8$

Graph : Use graphing device to graph the function $xy=8$ .

Graph : Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expression $xy=8$ which is required to graph.

Step 4: Press $\boxed{\text{GRAPH}}$ button to graph the function.

The graph is obtained as:

  

Interpretation : It is observed from the above graph that it is shows the hyperbola