Chapter 10.4, Problem 37AYU

To determine

To find: The equation of the hyperbola.

Answer to Problem 37AYU

Solution:

$x^2-y^2=1$

Explanation of Solution

Given:

The hyperbola

Formula used:

Center	Transverse axis	Foci	Vertices	Equation	Asymptotes
$(0,0)$	$x\text{-axis}$	$\left(\pm c,0\right)$	$\left(\pm a,0\right)$	$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $b^2=c^2-a^2$	$y=\frac{b}{a}x$ $y=-\frac{b}{a}x$

Calculation:

From the graph we see that,

Center: $(0,0)$.

Vertices: $\left(\pm 1,0\right)$; hence $a=1$.

Also, $\text{b = }1$ (from the graph).

$c²=b²+a²=1+1=2$; $c=\surd 2$

Foci: $\left(\pm c,0\right)=\left(\pm \sqrt{2},0\right)$.

Transverse Axis is the $x$ -axis.

Given that the asymptotes are $y=\pm x$.

Hence, the equation is

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$x²-y²=\text{ 1}$

The equation of the given graph is $x²-y²=1$.