# Hyperbola

Did you know that the orbit of a spacecraft can sometimes be a hyperbola?

A spacecraft can use the gravity of a planet to alter its path and propel it at high speed away from the planet and back out into space using a technique called "gravitational slingshot".

If this happens, then the path of the spacecraft is a **hyperbola**.

(Play with this at Gravity Freeplay)

## Definition

A hyperbola is two curves that are like infinite bows.

Looking at just one of the curves:

any point **P** is closer to **F** than to **G** by some constant amount

The other curve is a mirror image, and is closer to G than to F.

In other words, the distance from **P to F** is always less than the distance **P to G** by some constant amount. (And for the other curve **P to G** is always less than **P to F** by that constant amount.)

As a formula:

|PF − PG| = constant

- PF is the distance P to F
- PG is the distance P to G
- || is the absolute value function (makes any negative a positive)

Each bow is called a **branch** and F and G are each called a **focus**.

Have a try yourself:

Try moving point **P**: what do you notice about the lengths **PF** and **PG** ?

Also try putting point **P** on the other branch.

There are some other interesting things, too:

On the diagram you can see:

- an
**axis of symmetry**(that goes through each focus) - two
**vertices**(where each curve makes its sharpest turn) - the distance between the vertices (2a on the diagram) is the
**constant difference**between the lengths**PF**and**PG** - two
**asymptotes**which are not part of the hyperbola but show where the curve would go if continued indefinitely in each of the four directions

And, strictly speaking, there is also **another axis of symmetry** that goes down the middle and separates the two branches of the hyperbola.

## Conic SectionYou can also get a hyperbola when you slice through a double cone. The slice must be steeper than that for a parabola, but does not So the hyperbola is a conic section (a section of a cone). |

## Equation

By placing a hyperbola on an x-y graph (centered over the x-axis and y-axis), the equation of the curve is:

\frac{x^{2}}{a^{2}} − \frac{y^{2}}{b^{2}} = 1

Also:

One **vertex** is at (a, 0), and the other is at (−a, 0)

The **asymptotes** are the straight lines:

- y = (b/a)x
- y = −(b/a)x

(Note: the equation is similar to the equation of the ellipse: **x ^{2}/a^{2} + y^{2}/b^{2} = 1**, except for a "−" instead of a "+")

## Eccentricity

Any branch of a hyperbola can also be defined as a curve where the distances of any point from:

- a fixed point (the
**focus**), and - a fixed straight line (the
**directrix**) are always in the same ratio.

This ratio is called the eccentricity, and for a hyperbola it is always greater than 1.

The eccentricity (usually shown as the letter e) shows how "uncurvy" (varying from being a circle) the hyperbola is.

On this diagram:

- P is a point on the curve,
- F is the focus and
- N is the point on the directrix so that PN is perpendicular to the directrix.

The eccentricity is the ratio PF/PN, and has the formula:

e = \frac{√(a^{2}+b^{2})}{a}

Using "a" and "b" from the diagram above.

## Latus Rectum

The Latus Rectum is the line through the focus and parallel to the directrix. The length of the Latus Rectum is 2b |

## 1/x

The reciprocal function y = 1/x is a hyperbola!