# Foci of Hyperbola

The foci (singular: focus) of a hyperbola are two fixed points on its major axis, equidistant from the center. For any point on the hyperbola, the absolute difference between its distances to the two foci remains constant.

For a hyperbola given by the equation ${\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1}$, the foci are located at the coordinates (-c, 0) and (c, 0). Here, c is the distance between the center and the focus.

Thus, a graph of a hyperbola has two foci that are equidistant from the center.

Similarly, for any point on the hyperbola, the ratio of its distance from the foci to its distance from the directrix is always a constant called eccentricity, which is always greater than 1 (e > 1). Mathematically,  ${e=\dfrac{c}{a}}$, thus the coordinates of the foci are (-ae, 0) and (ae, 0).

Also, the midpoint of the foci is the center of the hyperbola

The foci, center (midpoint of foci), and vertices of a hyperbola are collinear and lie on the same axis.

## Formulas

Here are the formulas to calculate the foci for the different types of hyperbolas.

## Steps to Find Foci

The first step to finding the foci of a hyperbola is to identify whether the hyperbola is horizontal or vertical from its equation.

### In Horizontal Hyperbola

Let us find the foci for the hyperbola ${\dfrac{x^{2}}{36}-\dfrac{y^{2}}{9}=1}$

Identifying the Hyperbola

Since the given hyperbola ${\dfrac{x^{2}}{36}-\dfrac{y^{2}}{9}=1}$ is in the form ${\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1}$, it is a horizontal hyperbola.

Determining the Center of the Hyperbola and the Integers a and b

Comparing with the standard equation, we get

• The center of the hyperbola is at (0, 0)
• a = 6 and b = 3

Using the Formula c2 = a2 + b2

c2 = 62 + 32

⇒ c2 = 45

⇒ c = ${\sqrt{45}}$

Finding the Coordinates of the Foci

As we know, the foci of the hyperbola are (-c, 0) and (c, 0)

Thus, its coordinates are ${\left( -\sqrt{45},0\right)}$ and ${\left( \sqrt{45},0\right)}$

Problem – Finding the Foci of the Hyperbola given by the equation ${\dfrac{\left( x-h\right) ^{2}}{a^{2}}-\dfrac{\left( y-k\right) ^{2}}{b^{2}}=1}$

Find the foci of the hyperbola given by the equation
${\dfrac{\left( x+5\right) ^{2}}{81}-\dfrac{\left( y-7\right) ^{2}}{144}=1}$

Solution:

Since the given equation ${\dfrac{\left( x+5\right) ^{2}}{81}-\dfrac{\left( y-7\right) ^{2}}{144}=1}$ …..(i)
is in the form ${\dfrac{\left( x-h\right) ^{2}}{a^{2}}-\dfrac{\left( y-k\right) ^{2}}{b^{2}}=1}$ …..(ii), it is a horizontal hyperbola
On Comparing (i) and (ii), we get
The center of the hyperbola is at (h, k) = (-5, 7)
a = 9 and b = 12
Using the Formula c2 = a2 + b2
c2 = 92 + 122
⇒ c2 = 225
⇒ c = ${\sqrt{225}}$ = 15
Now, the foci of the hyperbola are (h – c, k) and (h + c, k)
⇒ (-5 – 15, 7) and (-5 + 15, 7)
Thus, the coordinates of the foci are (-20, 7) and (10, 7)

### In Vertical Hyperbola

Now, if we are given to find the foci of the hyperbola ${\dfrac{y^{2}}{25}-\dfrac{x^{2}}{144}=1}$, we will follow the following steps:

Identifying the Hyperbola

Since the given hyperbola ${\dfrac{y^{2}}{25}-\dfrac{x^{2}}{144}=1}$ is similar to its standard form ${\dfrac{y^{2}}{a^{2}}-\dfrac{x^{2}}{b^{2}}=1}$, it is a vertical hyperbola.

Determining the Center of the Hyperbola and the Integers a and b

Comparing with the standard equation, we get

• The center of the hyperbola is at (0, 0)
• a = 5 and b = 12

Using the Formula c2 = a2 + b2

c2 = 52 + 122

⇒ c2 = 169

⇒ c = ${\sqrt{169}}$ = 13

Finding the Coordinates of the Foci

As we know, the foci of the hyperbola are (0, c) and (0, -c)

Thus, its coordinates are (0, 13) and (0, -13)

Problem – Finding the Foci of the Hyperbola given by the equation ${\dfrac{\left( y-k\right) ^{2}}{a^{2}}-\dfrac{\left( x-h\right) ^{2}}{b^{2}}=1}$

Find the foci of the hyperbola given by the equation
${\dfrac{\left( y-2\right) ^{2}}{16}-\dfrac{\left( x+3\right) ^{2}}{64}=1}$

Solution:

Since the given hyperbola ${\dfrac{\left( y-2\right) ^{2}}{16}-\dfrac{\left( x+3\right) ^{2}}{64}=1}$ …..(i)
is in the form  ${\dfrac{\left( y-k\right) ^{2}}{a^{2}}-\dfrac{\left( x-h\right) ^{2}}{b^{2}}=1}$ …..(ii) it is a vertical hyperbola.
OnComparing (i) and (ii), we get
The center of the hyperbola is at (h, k) = (2, -3)
a = 4 and b = 8
Using the Formula c2 = a2 + b2
c2 = 42 + 82
⇒ c2 = 80
⇒ c = ${\sqrt{80}}$ = ${4\sqrt{5}}$
As we know, the foci of the hyperbola are (h, k + c) and (h, k – c)
⇒ ${\left( 2,-3+4\sqrt{5}\right)}$ and ${\left( 2,-3-4\sqrt{5}\right)}$
Thus, the coordinates of the foci are ${\left( 2,-3+4\sqrt{5}\right)}$ and ${\left( 2,-3-4\sqrt{5}\right)}$