Chapter 10, Problem 5RE

In Problems 1-10, identify each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci: if it is a hyperbola, give its center, vertices, foci, and asymptotes.

$4x^2-y^2\text{ = }8$

To determine

Each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci; if it is a hyperbola, give its center, vertices, foci, and asymptotes.

Answer to Problem 5RE

Hyperbola

Vertices: $(\pm 2\surd 2,0)$

Foci: $\left(\pm \surd 10, 0\right)$

Asymptotes: $y=\pm 2x$.

Explanation of Solution

Given:

$4x²-y²=8$

Formula used:

Equation	Center	Transverse axis	Foci	Vertices	Asymptotes
$\frac{x^2}{b^2}-\frac{y^2}{a^2}=1$ $b²=c²-a²$	$\left(0,0\right)$	$y\text{-axis}$	$\left(0,\pm c\right)$	$\left(0,\pm a\right)$	$y=\pm \frac{b}{a}x$

Calculation:

The equation $4x²-y²=8$ can be written as $\frac{x^2}{2}-\frac{y^2}{8}=1$.

The above equation is of the form $\frac{x^2}{2}-\frac{y^2}{8}=1$ and it represents a hyperbola.

$a=\sqrt{2},b=\surd 8=2\surd 2$

Center: $\left(0,0\right)$.

Vertices: $\left(\pm a,0\right)=(\pm 2\surd 2,0)$.

$c=\sqrt{a^2+b²}=\surd 10$

Foci: $\left(\pm c,0\right)=\left(\pm \sqrt{10},0\right)$.

Asymptotes: $y=\pm \frac{b}{a}x=\pm 2x$.