Hyperbolas on a Coordinate Plane

Hello everyone! I'm posting again, this time on hyperbolas.

The hyperbola is the set of all points in a plane such that the difference of the distances from two fixed points (the foci) is constant.

There are two different types.

Horizontal Hyperbola

Vertical Hyperbola

Transverse Axis Length= 2a

Conjugate Axis Length=2b

Note: The a² value is always the first denominator (not necessarily the larger) and the variable above the a² indicates if the transverse axis is horizontal or vertical.

Here is the standard form for both a Horizontal Hyperbola and Vertical Hyperbola. For the horizontal hyperbola, the horizontal tranverse axis opens sideways. The vertical hyperbola, the vertical tranverse axis opens up and down.

The General Form of a Hyperbola: Ax²-By²+Cx+Dy+E=0

You could tell that this is a hyperbola because of the negative sign between the A and B unlike the parabola where the sign instead is positive.

The equation for the asymptotes is:

Let's look at an example now.

1) Let's graph 9x² - 16y² = 144.

First, multiply each side of the equation by 1/144 to put it in standard form.

x² y²
-- - -- = 1
16 9

The direction of the hyperbola is horizontal. We now know that a = 4 and b = 3. The vertices are at (±4, 0). (Since we know the center is at the origin, we know the vertices are on the x axis.)

The easiest way to graph a hyperbola is to draw a box using the vertices and b, which is on the y-axis.

Draw the asymptotes through opposite corners of the box. Then draw the hyperbola.
The figure below is the graph of 9x² - 16y² = 144.

It's a lot more simple than it actually looks. Okay that's it! I didn't really do anything too fancy for this post so I hope that it's easy to understand. See you all tomorrow. :)