Chapter 11.3, Problem 43E

a.

To determine

To show:that the asymptotes of the hyperbola are perpendicular to each other.

Explanation of Solution

Given information:

The hyperbola equation is $x^2-y^2=5$ .

Proof:

First find the asymptotes of the hyperbola,

Since the $x^2$ -term is positive, the hyperbola has a horizontal transverse axis;

So the asymptote for the hyperbola is $y=\pm \frac{b}{a}x$ .

  $\begin{array}{l}{x}^2-y^2=5\\ \frac{x^2}{5}-\frac{y^2}{5}-1\text{ [divide by 5 and make standard form]}\end{array}$

Since here $a=\sqrt{5}\text{ and b=}\sqrt{5}$

So the asymptotes is,

  $\begin{array}{l}y=\pm \frac{\sqrt{5}}{\sqrt{5}}x\\ y=\pm x\end{array}$

The two asymptotes will be perpendicular is there slope product is -1.

Here the slope of $y=x$ asymptote is $1$ by comparing it by $y=mx+c$ where m is for slope.

And the slope of $y=-x$ asymptote is -1.

So the product of slope of the asymptotes is $1·\left(-1\right)=-1$ .

Hence the asymptotes are perpendicular.

b.

To determine

To find: the equation for the hyperbola.

Answer to Problem 43E

The equation of hyperbola is $x^2-y^2=\frac{c^2}{2}$ .

Explanation of Solution

The equation of hyperbola is $x^2-y^2=\frac{c^2}{2}$ .

Given information:

The hyperbola foci : $\left(\pm c,0\right)$

The asymptotes of hyperbola are perpendicular to each other.

Concept used: if the asymptotes of hyperbola are perpendicular to each other then $a^2=b^2$ .

Calculation: since from the given foci it is observed that the vertices of the hyperbola are on the x-axis, so the hyperbola has a horizontal transverse axis.

So the equation is of the form,

  $\frac{x^2}{{\left(a\right)}^2}-\frac{y^2}{b^2}=1$

to find the equation of hyperbola, substitute $a^2=b^2$ in the relation between $a,b,c$ .

use the relation,

  $\begin{array}{l}{c}^2=a^2+b^2\\ {\left(c\right)}^2={\left(a\right)}^2+a^2\text{ [substitute b by a]}\\ {\left(c\right)}^2=2{\left(a\right)}^2\text{ [add]}\\ a^2=\frac{c^2}{2}\text{ [divide]}\\ a^2=b^2\\ b^2=\frac{c^2}{2}\text{ [from the concept used]}\end{array}$

So the equation of hyperbola

  $\begin{array}{l}\frac{x^2}{{\left(a\right)}^2}-\frac{y^2}{b^2}=1\\ \frac{x^2}{\frac{c^2}{2}}-\frac{y^2}{\frac{c^2}{2}}=1\text{ [substitute a and b by}\frac{c^2}{2}\text{]}\\ x^2-y^2=\frac{c^2}{2}\text{ [simplify]}\end{array}$

Hence the equation of the hyperbola is $x^2-y^2=\frac{c^2}{2}$ .