Hyperbola: Formula, Foci, Transverse Axis

Hyperbola: Formula, Foci, Transverse Axis

How to solve the hyperbola problems: definition, formulas (equations, foci, transverse axis), proof, examples, and their solutions.

Definition

A hyperbola is the set of points whose difference of the distances from the foci is constant.

An hyperbola is the set of points
whose difference of the distances from the foci
is constant.
(= transverse axis, shown below)

The difference of the blue segments' lengths is constant.

Formula 1

The transverse axis is the segment whose endpoints are the vertices of the hyperbola.

The transverse axis is the segment
whose endpoints are the vertices of the hyperbola.
(blue, 2a)

If the vertices of the hyperbola are (+-a, 0), and if the foci are (+-c, 0), then the hyperbola is (x^2)/(a^2) - (y^2)/(b^2) = 1. And a^2 + b^2 = c^2.

If the vertices of the hyperbola are (±a, 0),
and if the foci are (±c, 0),

then the equation of the hyperbola is
x2/a2 - y2/b2 = 1.

And a2 + b2 = c2.

If the y2 term is (-),
(and if the right side is 1, not -1,)
the foci are on the x-axis.

So, if the y2 term is (-), use this formula.

The shape of a hyperbola is different
from the shape of a parabola.

Proof: Formula 1

Hyperbola: Proof of the Formula 1

Set [distance between (x, y) and (-c, 0)]
- [distance between (x, y) and (c, 0)]
= ±2a.

Move the second radical term to the right side.

Square both sides.

Square of a sum

Cancel the gray terms.

Then arrange the terms
so that only the remaining radical term
is on the left side.

Divide both sides by 4.

Square both sides.

Square of a sum

Cancel the gray terms.

Make (a2 - c2) by factoring.

Then put -b2
into a2 - c2.

Divide both sides by a2(-b2).

Then the equaition of the hyperbola is
x2/a2 - y2/b2 = 1.

And the relationship between a, b, and c is
a2 + b2 = c2.

Example 1

The equation of a hyperbola is given below. (x^2)/9 - (y^2)/16 = 1 1. Find the length of the transverse axes. 2. Find the coordinates of the foci.

To find a, b, and c,
change the equation into the standard form.

x2/32 - y2/42 = 1

Then a = 3 and b = 4.

y2 term is (-).

So (transverse axis) = 2⋅3 = 6.

Use a2 + b2 = c2
to find c.

c2 = a2 + b2
= 32 + 42
c = ±5

So the foci are (±5, 0).

This is the graph of the hyperbola
you've solved above.

Example 2

Write an equation of the hyperbola described below. foci: (-1, -1), (5, -1), transverse axis: 2

Roughly draw the given conditions.

Foci: (-1, -1), (5, -1)

The distance between the foci is 2|c|.

The foci are located horizontally.
So the transverse axis is 2a = 2.

The distance between the foci (-1, -1), (5, -1)
is 2|c|.

So 2|c| = 5 - (-1) = 6.

c = ±3

The foci are not (±3, 0).
So the hyperbola is under a translation.

Translation of a point

So use c = 3
to find the changes of x and y:
(5, -1) = (3 + 2, -1).

So the hyperbola is under the translation
(x, y) → (x + 2, y - 1).

The transverse axis is 2a = 2.
So a = 1.

Use a2 + b2 = c2
to find b.

12 + b2 = 32

b2 = 8

Write the equation of the hyperbola
using the above conditions.

(x - 2)2/12 - (y + 1)2/8 = 1

Translation of a function

See how (x - 2)2/12 - (y + 1)2/8 = 1 looks like.

The graph is determined by
its foci (+3 + 2, -1), (-3 + 2, -1)
and its transverse axis 2⋅1.

Formula 2

If the vertices of the hyperbola are (0, +-b), then the transverse axis is vertical.

If the vertices of the hyperbola are (0, ±b),
then the transverse axis is vertical.
(green, 2b)

If the vertices of the hyperbola are (0, +-b), and if the foci are (0, +-c), then the hyperbola is (y^2)/(b^2) - (x^2)/(a^2) = 1. And a^2 + b^2 = c^2.

If the vertices of the hyperbola are (0, ±b),
and if the foci are (0, ±c),

then the equation of the hyperbola is
y2/b2 - x2/a2 = 1.

And a2 + b2 = c2.

If the x2 term is (-), the foci are on the y-axis.
So, if the x2 term is (-), use this formula.

Proof: Formula 2

Hyperbola: Proof of the Formula 2

Set [distance between (x, y) and (0, -c)]
- [distance between (x, y) and (0, c)]
= ±2b.

Move the second radical term to the right side.

Square both sides.

Square of a sum

Cancel the gray terms.

Then arrange the terms
so that only the remaining radical term
is on the left side.

Divide both sides by 4.

Square both sides.

Square of a sum

Cancel the gray terms.

Make (b2 - c2) by factoring.

Then put -a2
into b2 - c2.

Switch the left side terms.

And divide both sides by b2(-a2).

Then the equaition of the hyperbola is
y2/b2 - x2/a2 = 1.

And the relationship between a, b, and c is
a2 + b2 = c2.

Example 3

The equation of a hyperbola is given below. x^2 - y^2 = -4 1. Find the length of the transverse axes. 2. Find the coordinates of the foci.

To find a, b, and c,
change the equation into the standard form.

y2/22 - x2/22 = 1

Then a = b = 2.

x2 term is (-).

So (transverse axis) = 2⋅2 = 4.
(2 from b)

Use a2 + b2 = c2
to find c.

c2 = a2 + b2
= 22 + 22
c = ±2√2

So the foci are (0, ±2√2).

This is the graph of the hyperbola
you've solved above.