Hyperbola

In a hyperbola, the difference between the distances from two fixed points is constant. These fixed points are the foci of the hyperbola. These points are located at (c,0) and (-c,0).

The standard form is: x^{2}/a^{2} – y^{2}/b^{2} = 1

Also: a^{2} + b^{2} = c^{2}

Note that this looks just like the ellipse equation except for the minus sign.

To sketch a hyperbola, you should first sketch its asymptotes. The asymptotes are seen below.

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EXAMPLE: Sketch x^{2} – 4y^{2} – 16 = 0

Solution: First rearrange this to look like the standard form. How?

Add 16 to both sides. Then divide by 16. You get:

x^{2}/16 – y^{2}/4 = 1

So a = 4 and b = 2. Now connect the points (4,2), (-4,-2) and (4,-2), (-4,2). These two lines are the asymptotes.

Calculate c: c^{2} = a^{2} + b^{2} = 4^{2} + 2^{2}

c = 4.472

So the foci are (4.472, 0) and (-4.472, 0).

Now we can sketch the hyperbola.

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Click here for practice problems.