How do you find the coordinates of the vertices, foci, and the equation of the asymptotes for the hyperbola $x^{2} - y^{2} = 4$ $x^2-y^2=4$ ?

Question

How do you find the coordinates of the vertices, foci, and the equation of the asymptotes for the hyperbola $x^{2} - y^{2} = 4$ $x^2-y^2=4$ ?

1 Answer

Impact of this question

2420 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-06-23T20:23:41

Vertices: $(\pm 2, 0)$ $(+-2, 0)$
Foci: $(\pm 2 \sqrt{2}, 0)$ $(+-2sqrt2, 0)$
Asymptotes: $y = \pm x$ $y=+-x$

Explanation:

First, we need to have our equation in standard form:

$\frac{x^{2}}{4} - \frac{y^{2}}{4} = 1$ $x^2/4-y^2/4=1$

That's better! To get to our answer, we need to know that our hyperbola is a horizontal hyperbola, which means that the vertices and foci will be have the same y-coordinate as the center.

Next, we need to know our the values of $a$ $a$ , $b$ $b$ , and $c$ $c$ . These help us find out the dimensions of our graph. The formula for a horizontal hyperbola is

$\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$ $(x-h)^2/a^2-(y-k)^2/b^2=1$

In our case, the $h$ $h$ and $k$ $k$ values don't exist, so our center is at the origin. $a^{2}$ $a^2$ and $b^{2} = 4$ $b^2 =4$ , so both $a$ $a$ and $b = 2$ $b = 2$ . To find $c$ $c$ , we will use an equation you've probably seen before:

$a^{2} + b^{2} = c^{2}$ $a^2+b^2=c^2$
$4 + 4 = c^{2}$ $4+4=c^2$
$8 = c^{2}$ $8=c^2$
$2 \sqrt{2} = c$ $2sqrt2=c$

To find our vertices and foci, we move out a certain amount of units in both directions: $a$ $a$ for the vertices and $c$ $c$ for the foci.

Vertices: $(\pm 2, 0)$ $(+-2, 0)$
Foci: $(\pm 2 \sqrt{2}, 0)$ $(+-2sqrt2, 0)$

Finally, let's get the equations of the hyperbola's asymptotes. The equations are $y = \pm \frac{b}{a} x$ $y=+-b/a x$ for a horizontal hyperbola at $(0, 0)$ $(0,0)$ . Therefore, our equations are $y = \pm x$ $y=+-x$ .

Science

Math

Humanities

... and beyond

How do you find the coordinates of the vertices, foci, and the equation of the asymptotes for the hyperbola $x^{2} - y^{2} = 4$ $x^2-y^2=4$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you find the coordinates of the vertices, foci, and the equation of the asymptotes for the hyperbola x2−y2=4x^2-y^2=4?

1 Answer

Explanation:

Related questions

Impact of this question

How do you find the coordinates of the vertices, foci, and the equation of the asymptotes for the hyperbola $x^{2} - y^{2} = 4$ $x^2-y^2=4$ ?