Cooling towers for a nuclear power plant have a hyperbolic cross-section.
[Image source.]
How do we create a hyperbola?
Take 2 fixed points A and B and let them be 4a units apart. Now, take half of that distance (i.e. 2a units).
Now, move along a curve such that from any point on the curve,
(distance to A) − (distance to B) = 2a units.
The curve that results is called a hyperbola. There are two parts to the curve.
Example 1
Let the distance between our points A and B be 4 cm. For convenience, let's place our fixed points A and B on the number line at (0, 2) and (0, -2), so they are 4 units apart. In this case, a = 1 cm and 2a = 2 cm.
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More Forms of the Equation of a Hyperbola
Even More Forms of the Equation of a Hyperbola
Applications of Hyperbolas
See some interesting Applications [external site]
Now we start tracing out a curve such that P is a point on the curve and the (distance PB) minus (distance PA) is 2 cm. We start at (0, 1).
If we continue, we obtain:
Now, continuing our curve on the left side of the axis gives us:
We also have another part of the hyperbola on the opposite side of the x-axis, this time using:
distance PA − distance PB = 2
We observe that the curves become almost straight near the extremities. In fact, the lines y = x and y = -x (the red dotted lines below) are asymptotes:
[An asymptote is a line that forms a "barrier" to a curve. The curve gets closer and closer to an asymptote, but does not touch it.]
In Example 1, the points (0, 1) and (0, -1) are called the vertices of the hyperbola.
The Equation of a Hyperbola
For the hyperbola with a = 1 that we graphed above in Example 1, the equation is given by:
y2 − x2 = 1
Notice that it is not a function, since for each x-value, there are two y-values.
We call this example a "north-south" opening hyperbola.
East-West Opening Hyperbola
By reversing the x- and y-variables in our example above, we obtain the following equation.
Example 2
x2 − y2 = 1
This gives us an "East-West" opening hyperbola, as follows. Our curve passes through -1 and 1 on the x-axis and once again, the asymptotes are the lines y = x and y = -x.
Technical Definition of a Hyperbola
A hyperbola is the locus of points where the difference in the distance to two fixed foci is constant.
This technical definition is one way of describing what we were doing in Example 1, above.
Application of Hyperbolas
Throw 2 stones in a pond. The resulting concentric ripples meet in a hyperbola shape.
More Forms of the Equation of a Hyperbola
There are a few different formulas for a hyperbola.
Considering the hyperbola with centre (0, 0), the equation is either:
1. For a north-south opening hyperbola:
The slopes of the asymptotes are given by:
2. For an east-west opening hyperbola:
The slopes of the asymptotes are given by:
In the examples given above, both a and b were equal to 1, so the slopes of the asymptotes were simply ± 1 and our asymptotes were the lines y = x and y = -x.
What effect does it have if we change a and b?
Example 3
Sketch the hyperbola
Even More Forms of the Equation of a Hyperbola
1. Possibly the simplest equation of a hyperbola is given in the following example.
Example 4 - Equilateral Hyperbola
xy = 1
This is known as the equilateral or rectangular hyperbola.
Notice that this hyperbola is a "north-east, south-west" opening hyperbola. Compared to the other hyperbolas we have seen so far, the axes of the hyperbola have been rotated by 45°. Also, the asymptotes are the x- and y-axes.
2. Our hyperbola may not be centred on (0, 0). In this case, we use the following formulas:
For a "north-south" opening hyperbola with centre (h, k), we have:
For an "east-west" opening hyperbola with centre (h, k), we have:
Example 5 - Hyperbola with Axes Shifted
Sketch the hyperbola
3. We could expand our equations for the hyperbola into the following form:
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 (such that B2 > 4AC)
In the earlier examples on this page, there was no xy-term involved. As we saw in Example 4, if we do have an xy-term, it has the effect of rotating the axes. We no longer have "north-south" or "east-west" opening arms - they could open in any direction.
Example 6 - Hyperbola with Shifted and Rotated Axes
The graph of the hyperbola x2 + 5xy − 2y2 + 3x + 2y + 1 = 0 is as follows:
We see that the axes of the hyperbola have been rotated and have been shifted from (0, 0).
[Further analysis is beyond the scope of this section. ]
Exercise
Sketch the hyperbola
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