# Understanding Hyperbolas: Equations, Characteristics, and Examples

## Hyperbola as a Geometric Locus

The hyperbola is defined as the geometric locus of all points in a plane where the difference of distances from two fixed points (called foci) is constant.

The equation of a hyperbola is similar to that of an ellipse, but instead of the sum of distances, it involves the difference of distances.

## Ordinary Form of the Hyperbola Equation

### Hyperbola with Horizontal Focal Axis and Center at the Origin (0, 0)

The equation for this hyperbola is: *x*^{2}/*a*^{2} – *y*^{2}/*b*^{2} = 1

**Characteristics:**

- Symmetric with respect to the
*x*and*y*axes. - Foci: F (±
*c*, 0) - Vertices: V (±
*a*, 0) - Minor Axis Endpoints: B (0, ±
*b*) - Center: Origin (0, 0)
- Transverse Axis Length: 2
*a* - Conjugate Axis Length: 2
*b* - Focal Axis Length: 2
*c* - Eccentricity:
*e*=*c*/*a*(where*c*^{2}=*a*^{2}+*b*^{2}) - Latus Rectum: 2
*b*^{2}/*a* - Directrix:
*x*= ±*a*^{2}/*c* - Asymptotes:
*y*= ±(*b*/*a*)*x*

### Hyperbola with Vertical Focal Axis and Center at the Origin (0, 0)

The equation for this hyperbola is: *y*^{2}/*a*^{2} – *x*^{2}/*b*^{2} = 1

**Characteristics:**

- Foci: F (0, ±
*c*) - Vertices: V (0, ±
*a*) - Minor Axis Endpoints: B (±
*b*, 0) - Center: Origin (0, 0)
- Transverse Axis Length: 2
*a* - Conjugate Axis Length: 2
*b* - Focal Axis Length: 2
*c* - Eccentricity:
*e*=*c*/*a*(where*c*^{2}=*a*^{2}+*b*^{2}) - Latus Rectum: 2
*b*^{2}/*a* - Directrix:
*y*= ±*a*^{2}/*c* - Asymptotes:
*y*= ±(*a*/*b*)*x*

### Example: Horizontal Hyperbola

Given the equation *x*^{2}/9 – *y*^{2}/25 = 1, determine:

- Equation of the asymptotes
- Coordinates of the foci
- Coordinates of the vertices
- Length of the latus rectum

**Solution:**

- Asymptotes:
*y*= ±(5/3)*x* - Foci: F (±√34, 0)
- Vertices: V (±3, 0)
- Latus Rectum: 50/3

### Hyperbola with Center (h, k)

#### Horizontal Focal Axis

Equation: (*x* – *h*)^{2}/*a*^{2} – (*y* – *k*)^{2}/*b*^{2} = 1

#### Vertical Focal Axis

Equation: (*y* – *k*)^{2}/*a*^{2} – (*x* – *h*)^{2}/*b*^{2} = 1

### Example: Finding Center, Vertices, and Foci

Given the equation (*x* – 0)^{2}/4 – (*y* – 1)^{2}/9 = 1, find the center, coordinates of the vertices, and foci.

**Solution:**

- Center: C (0, 1)
- Vertices: V (-2, 1) and V (2, 1)
- Foci: F (-√13, 1) and F (√13, 1)

### Example: Finding the Equation of a Hyperbola

The center of a hyperbola is at point C (-2, 5) and its latus rectum is 4. If the length of its transverse axis is 8 and it is parallel to the *x* axis, determine its equation.

**Solution:**

Using the given information and the standard equation for a horizontal hyperbola, we can derive the equation: (*x* + 2)^{2}/16 – (*y* – 5)^{2}/4 = 1