6. The Hyperbola

hyperbolic cooling towers
Cooling towers for a nuclear power plant have a hyperbolic cross-section.
[Image source.]

A hyperbola is a pair of symmetrical open curves. It is what we get when we slice a pair of vertical joined cones with a vertical plane.

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.

Let's see how this works with some examples.

Example 1

Let the distance between our points A and B be 4 cm. For convenience in our first example, 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.

On this page...

The Equation of a Hyperbola

East-West Opening Hyperbola

Definition of a Hyperbola

More Forms of the Equation of a Hyperbola

Even More Forms of the Equation of a Hyperbola

Applications of Hyperbolas

  • Navigation: Ship's navigators can plot their position by comparing GPS signals from different satellites. The technique involves hyperbolas.
  • Physics: The movement of objects in space and of subatomic particles trace out hyperbolas in certain situations.
  • Sundials: Historically, sundials made use of hyperbolas. Place a stick in the ground and trace out the path made by the shadow of the tip, and you'll get a hyperbola.
  • Construction: Nuclear power plant smoke stacks have a hyperbolic cross section as illustrated above. Such 3-dimensional objects are called hyperboloids.

hyperbola basics - 2 points

Now we start tracing out a curve such that P is a point on the curve, and:

distance PB − distance PA `= 2` cm.

We start at `(0, 1)`.

Shown below is one of the points P, such that `PB − PA = 2`.

constructing hyperbola 2

If we continue, we obtain the blue curve:

constructing hyperbola 3

Now, continuing our curve on the left side of the axis gives us the following:

constructing hyperbola 4

We also have another part of the hyperbola on the opposite side of the x-axis, this time using:

distance PA − distance PB = 2

Once again a typical point P is shown, and we can see from the lengths given that `PA − PB = 2`.

constructing hyperbola 5

We observe that the curves become almost straight near the extremities. In fact, the lines `y=x/sqrt3` and `y=-x/sqrt3` (the red dotted lines below) are asymptotes:

hyperbola and 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, while the points `(0, 2)` and `(0, -2)` are the foci (or focuses) of the hyperbola.

The equation of our hyperbola

For the hyperbola with a = 1 that we graphed above in Example 1, the equation is given by:

`y^2-x^2/3=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.

Where did this hyperbola equation come from?

The equation follows from the distance formula and the requirement (in this example) that distance PB − distance PA = 2.

Here's the proof.

In general

For the hyperbola with focal distance 4a (distance between the 2 foci), and passing through the y-axis at (0, c) and (0, −c), we define

b2 = c2a2

Applying the distance formula for the general case, in a similar fashion to the above example, we obtain the general form for a north-south hyperbola:

`y^2/a^2-x^2/b^2=1`

Example 2

Here's another example of an "north-south" hyperbola.

It's equation is:

y2x2 = 1

hyperbola with asymptotes

Similar to Example 1, this hyperbola passes through `1` and `−1` on the y-axis, but it has a different equation and a slightly different shape (and different asymptotes). Where are the 2 foci for this hyperbola? We need to find the value of c.

By inspection (of the equation of this hyperbola), we can see a = 1 and b = 1. Using the formula given above, we have:

b2 = c2a2

So

12 = c2 − 12

c2 = 2

c = ±√2

So the points A and B (the foci) for this hyperbola are at A (0, √2) and B (0, −√2).

East-West Opening Hyperbola

By reversing the x- and y-variables in our second example above, we obtain the following equation.

Example 3

x2y2 = 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.

east-west hyperbola

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.

Hyperbolas in Nature

ripples in pond - hyperbola

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:

`y^2/a^2-x^2/b^2=1`

The slopes of the asymptotes are given by:

`+-a/b`

2. For an east-west opening hyperbola:

`x^2/a^2-y^2/b^2=1`

The slopes of the asymptotes are given by:

`+-b/a`

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 4

Sketch the hyperbola

`y^2/25-x^2/4=1`

Even More Forms of the Equation of a Hyperbola

(1) Possibly the simplest equation of a hyperbola is given in the following example.

Example 5 - Equilateral Hyperbola

xy = 1

This is known as the equilateral or rectangular hyperbola.

xy = 1 rectangula 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.

Hyperbola with axis not at the Origin

(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:

`((y-k)^2)/a^2-((x-h)^2)/b^2=1`

For an "east-west" opening hyperbola with centre (h, k), we have:

`((x-h)^2)/a^2-((y-k)^2)/b^2=1`

Example 6 - Hyperbola with Axes Shifted

Sketch the hyperbola

`((x-2)^2)/36-((y+3)^2)/64=1`

3. We could expand our equations for the hyperbola into the following form:

`Ax^2+ Bxy + Cy^2+ Dx + Ey + F = 0` (such that `B^2>4AC`)

In the earlier examples on this page, there was no xy-term involved. As we saw in Example 5, 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 7 - Hyperbola with Shifted and Rotated Axes

The graph of the hyperbola x2 + 5xy − 2y2 + 3x + 2y + 1 = 0 is as follows:

hyperbola with rotated axes

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

`x^2/9-y^2/16=1`

Conic section: Hyperbola

How can we obtain a hyperbola from slicing a cone?

We start with a double cone (2 right circular cones placed apex to apex):

double cone

When we slice the 2 cones, we get a hyperbola, as shown.

hyperbola conic section

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