How to Find Foci of a Hyperbola?

To find the foci of a hyperbola, you primarily need to determine the values of 'a', 'b', and then use the fundamental relationship $c^2 = a^2 + b^2$ to find 'c'. Once 'c' is known, the foci are located 'c' units along the transverse axis from the center of the hyperbola.

Understanding Hyperbola Foci

The foci (plural of focus) are two fixed points inside a hyperbola that are crucial to its definition. For any point on the hyperbola, the absolute difference of its distances to the two foci is a constant value. These points also help define the shape and orientation of the hyperbola.

The Key Formula: Finding 'c'

The distance from the center of the hyperbola to each focus is denoted by 'c'. This value 'c' is directly related to 'a' and 'b', which are derived from the hyperbola's standard equation:

• 'a' represents the distance from the center to a vertex along the transverse (main) axis.
• 'b' represents the distance from the center to a co-vertex along the conjugate axis.

The relationship between 'a', 'b', and 'c' for a hyperbola is given by:

$c^2 = a^2 + b^2$

Once 'c' is calculated, you can determine the exact coordinates of the foci.

Step-by-Step Guide to Locating Foci

Follow these steps to find the foci of any given hyperbola:

1. Identify the Center (h, k)

The standard form of a hyperbola equation helps identify its center.

• If the equation is in the form $\frac{(x-h)^2}{A} - \frac{(y-k)^2}{B} = 1$ or $\frac{(y-k)^2}{A} - \frac{(x-h)^2}{B} = 1$, the center is $(h, k)$.
• If the hyperbola is centered at the origin, its center is $(0, 0)$, meaning $h=0$ and $k=0$.

2. Determine 'a' and 'b' from the Standard Equation

From the standard form of the hyperbola equation:

• The value under the positive term is $a^2$. So, find 'a' by taking the square root of that value.
• The value under the negative term is $b^2$. So, find 'b' by taking the square root of that value.

3. Calculate 'c' using the Focal Relationship

Substitute the values of $a^2$ and $b^2$ into the formula $c^2 = a^2 + b^2$ and solve for 'c'. Remember that 'c' must be a positive value, as it represents a distance.

4. Identify the Transverse Axis Orientation

The orientation tells you whether the hyperbola opens horizontally (along the x-axis) or vertically (along the y-axis), which in turn dictates where the foci lie:

• Horizontal Hyperbola: The $x^2$ term (or $(x-h)^2$ term) is positive. The transverse axis is parallel to the x-axis. The foci will have coordinates $(h \pm c, k)$.
• Vertical Hyperbola: The $y^2$ term (or $(y-k)^2$ term) is positive. The transverse axis is parallel to the y-axis. The foci will have coordinates $(h, k \pm c)$.

5. Apply 'c' to the Center Coordinates

Add and subtract the value of 'c' from the appropriate coordinate of the center $(h, k)$ based on the transverse axis orientation:

• For a horizontal hyperbola, the foci are at $(h - c, k)$ and $(h + c, k)$.
• For a vertical hyperbola, the foci are at $(h, k - c)$ and $(h, k + c)$.

Examples of Finding Foci

Let's illustrate with a couple of examples.

Example 1: Horizontal Hyperbola Centered at the Origin

Consider a hyperbola where $a^2 = 9$ and $b^2 = 16$, centered at $(0, 0)$.

1. Center: $(h, k) = (0, 0)$.
2. Determine 'a' and 'b': $a^2 = 9 \implies a = 3$; $b^2 = 16 \implies b = 4$.
3. Calculate 'c':
   $c^2 = a^2 + b^2$
   $c^2 = 9 + 16$
   $c^2 = 25$
   $c = \sqrt{25}$
   $c = 5$
4. Orientation: Since the $x^2$ term would be positive (e.g., $\frac{x^2}{9} - \frac{y^2}{16} = 1$), it's a horizontal hyperbola.
5. Locate Foci: Counting 5 units to the left and right of the center $(0,0)$, the coordinates of the foci are $(-5, 0)$ and $(5, 0)$.

Example 2: Vertical Hyperbola with a Shifted Center

Suppose a hyperbola has the equation $\frac{(y+1)^2}{25} - \frac{(x-2)^2}{144} = 1$.

1. Center: $(h, k) = (2, -1)$.
2. Determine 'a' and 'b':
   The positive term is $\frac{(y+1)^2}{25}$, so $a^2 = 25 \implies a = 5$.
   The negative term is $\frac{(x-2)^2}{144}$, so $b^2 = 144 \implies b = 12$.
3. Calculate 'c':
   $c^2 = a^2 + b^2$
   $c^2 = 25 + 144$
   $c^2 = 169$
   $c = \sqrt{169}$
   $c = 13$
4. Orientation: The $(y+1)^2$ term is positive, so it's a vertical hyperbola.
5. Locate Foci: The foci will be 'c' units above and below the center $(2, -1)$.
   Foci are $(h, k \pm c) = (2, -1 \pm 13)$.
   The coordinates of the foci are $(2, -1 - 13) = (2, -14)$ and $(2, -1 + 13) = (2, 12)$.

Standard Forms and Foci Location Summary Table

The following table summarizes the standard forms of hyperbola equations and how to locate their foci:

By understanding these relationships and following the methodical steps, you can accurately determine the foci of any hyperbola.