Hyperbola: Equation
How to use the hyperbola equation to find the transverse axis and the foci (and vics versa): definition, formula, 6 examples, and their solutions.
Definition
A hyperbola is the set of points
whose difference of the distances from the foci
is constant.
|PF - PF'| = (constant)
FormulaHorizontal Hyperbola
This is the graph of the hyperbola
x2/a2 - y2/b2 = 1.
The x2 term, x2/a2, is (+).
Then this is a horizontal hyperbola.
To show that
the hyperbola is a horizontal hyperbola,
we write x2 term first.
The transverse axis is
the distance between the vertices.
The transverse axis is 2a.
For x2/a2 - y2/b2 = 1,
the foci are (c, 0) and (-c, 0).
a, b, and c satisfy
a2 + b2 = c2.
Unlike the ellipse formula,
a2 - b2 = c2,
the middle sign of the left side is (+).
ExampleTransverse Axis
9 = 32
16 = 42
x2/32 - y2/42 = 1
The x2 term is (+).
Then the transverse axis is
2⋅3.
2⋅3 = 6
So 6 is the answer.
ExampleFoci
You just found that
the given hyperbola is
x2/32 - y2/42 = 1.
x2/32 - y2/42 = 1
a = 3
b = 4
Then
c2 = 32 + 42.
32 = 9
+42 = +16
9 + 16 = 25
c2 = 25
Then c = √25.
Think the sign of the c plus.
25 = 52
√52 = 5
Square Root
c = 5
See x2/32 - y2/42 = 1.
The x2 term, x2/32, is (+).
Then the hyperbola is a horizontal hyperbola.
So the foci are
(5, 0) and (-5, 0).
So
(5, 0), (-5, 0)
is the answer.
ExampleHyperbola Equation
The foci are (3, 0) and (-3, 0).
So draw a horizontal hyperbola like this.
And draw the foci (3, 0) and (-3, 0).
Then c = 3.
The transverse axis is
the distance between the vertices.
It's 4.
So 2a = 4.
Divide both sides by 2.
Then a = 2.
a = 2
The foci are (3, 0) and (-3, 0).
So c = 3.
Then
22 + b2 = 32.
22 = 4
32 = 9
Move 4 to the right side.
Then b2 = 5.
Instead of finding b,
use b2 = 5
to write the hyperbola equation.
The hyperbola is a horizontal hyperbola.
So write the x2 term first.
a = 2
b2 = 5
Then the hyperbola is
x2/22 - y2/5 = 1.
22 = 4
So
x2/4 - y2/5 = 1
is the answer.
FormulaVertical Hyperbola
This is the graph of the hyperbola
y2/a2 - x2/b2 = 1.
The y2 term, y2/a2, is (+).
Then this is a vertical hyperbola.
To show that
the hyperbola is a vertical hyperbola,
we write y2 term first.
The transverse axis is 2a.
For y2/a2 - x2/b2 = 1,
the foci are (0, c) and (0, -c).
a, b, and c satisfy
a2 + b2 = c2.
ExampleTransverse Axis
To make the right side 1,
divide both sides by 4.
4 = 22
-x2
= -x2/1
= -x2/12
y2/22 - x2/12 = 1
The y2 term is (+).
Then the transverse axis is
2⋅2.
2⋅2 = 4
So 4 is the answer.
ExampleFoci
You just found that
the given hyperbola is
y2/22 - x2/12 = 1.
y2/22 - x2/12 = 1
a = 2
b = 1
Then
c2 = 22 + 12.
22 = 4
+12 = +1
4 + 1 = 5
c2 = 5
Then c = √5.
Think the sign of the c plus.
c = √5
See y2/22 - x2/12 = 1.
The y2 term, y2/22, is (+).
Then the hyperbola is a vertical hyperbola.
So the foci are
(0, √5) and (0, -√5).
So
(0, √5), (0, -√5)
is the answer.
ExampleHyperbola Equation
The foci are (0, 5) and (0, -5).
So draw a vertical hyperbola like this.
And draw the foci (0, 5) and (0, -5).
Then c = 5.
The transverse axis is
the distance between the vertices.
It's 6.
So 2a = 6.
Divide both sides by 2.
Then a = 3.
a = 3
The foci are (0, 5) and (0, -5).
So c = 5.
Then
32 + b2 = 52.
32 = 9
52 = 25
Move 9 to the right side.
Then b2 = 16.
Instead of finding b,
use b2 = 16
to write the hyperbola equation.
The hyperbola is a vertical hyperbola.
So write the y2 term first.
a = 3
b2 = 16
Then the hyperbola is
y2/32 - x2/16 = 1.
32 = 9
Then
y2/9 - x2/16 = 1
is the answer.