Hyperbola: Formula

Hyperbola: Formula

How to write the equation of a hyperbola by using its foci and transverse axis: formulas, examples, and their solutions.

Formula: x2/a2 - y2/b2 = 1

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.

Example 1: Foci (3, 0) and (-3, 0), Transverse Axis 4, Hyperbola?

Write an equation of the hyperbola whose foci and transverse axis are below. Foci: (3, 0), (-3, 0). Transverse axis: 4

The foci are (3, 0) and (-3, 0).

Then c = 3.

The transverse axis is 4.

And the y values of the foci are the same.
So the transverse axis is 2a.

So 2a = 4.

Hyperbola - Transverse axis

Divide both sides by 2.

Then a = 2.

a = 2
c = 3

So 22 + b2 = 32.

22 = 4

32 = 9

Move 4 to the right side.

Then b2 = 5.

Instead of finding the value of b,
directly use b2 = 5
to write the equation of the hyperbola.

a = 2
b2 = 5

So the equation of the hyperbola is
x2/22 - y2/5 = 1.

22 = 4

So [x2/4 - y2/5 = 1] is the answer.

This is the graph of x2/4 - y2/5 = 1.

Its foci are (3, 0) and (-3, 0).

And its transverse axis is 2⋅2 = 4.

Example 2: Foci (-3, 2) and (5, 2), Transverse Axis 6, Hyperbola?

Write an equation of the hyperbola whose foci and transverse axis are below. Foci: (-3, 2), (5, 2). Transverse axis: 6

Lightly draw the given conditions.

It says the foci are (-3, 2) and (5, 2).
And the transverse axis is 6.

So draw a hyperbola like this.

The foci are (-3, 2) and (5, 2).
And the distance between the foci is 2c.

So 2c = 5 - (-3).

-(-3) = +3

5 + 3 = 8

Divide both sides by 2.

Then c = 4.

c = 4
And the y values of the foci are the same.

So the original foci are (-4, 0) and (4, 0).

But the given foci are (-3, 2) and (5, 2).

So the foci are under a translation.

Use (4, 0) and (5, 2) to find the translation:
(5, 2) = (4 + 1, 0 + 2).

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

Translation of a point

Next, it says the transverse axis is 6.

And the y values of the foci are the same.
So the transverse axis is 2a.

So 2a = 6.

Divide both sides by 2.

Then a = 3.

a = 3
c = 4

So 32 + b2 = 42.

32 = 9

42 = 16

Move 9 to the right side.

Then b2 = 7.

Instead of finding the value of b,
directly use b2 = 7
to write the equation of the hyperbola.

a = 3
b2 = 7

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

So the equation of the hyperbola is
(x - 1)2/32 - (y - 2)2/7 = 1.

Translation of a function

32 = 9

So (x - 1)2/9 - (y - 2)2/7 = 1.

This is the graph of (x - 1)2/9 - (y - 2)2/7 = 1.

Its foci are (-4 + 1, 0 + 2) = (-3, 2)
and (4 + 1, 0 + 2) = (5, 2).

And its transverse axis is 2⋅3 = 6.

Formula: y2/b2 - x2/a2 = 1

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.

Example 3: Foci (0, 2) and (0, -2), Transverse Axis 2, Hyperbola?

Write an equation of the hyperbola whose foci and transverse axis are below. Foci: (0, 2), (0, -2). Transverse axis: 2

The foci are (0, 2) and (0, -2).

Then c = 2.

The transverse axis is 2.

And the x values of the foci are the same.
So the transverse axis is 2b.

So 2b = 2.

Hyperbola - Transverse axis

Divide both sides by 2.

Then b = 1.

b = 1
c = 2

So a2 + 12 = 22.

12 = 1

22 = 4

Move +1 to the right side.

Then a2 = 3.

Instead of finding the value of a,
directly use a2 = 3
to write the equation of the hyperbola.

a2 = 3
b = 1

So the equation of the hyperbola is
y2/12 - x2/3 = 1.

12 = 1

So [y2 - x2/3 = 1] is the answer.

This is the graph of y2 - x2/3 = 1.

Its foci are (0, 2) and (0, -2).

And its transverse axis is 2⋅1 = 2.