# Understanding the Ellipse: Equations, Characteristics, and Examples

## Ellipse as a Geometric Locus

The **ellipse** is defined as the geometrical locus formed by the set of all points in the plane, where the sum of the distances from two fixed points, known as the **foci** (plural of focus) of the ellipse, is a constant.

### Characteristics and Elements of the Ellipse

* V* and

*are the vertices of the ellipse and are also known as the endpoints of the major axis.*

**V’*** F* and

*are the foci of the ellipse.*

**F’*** B* and

*are the endpoints of the minor axis.*

**B’****Segment** is known as the major axis of the ellipse.

**Segment** is known as the minor axis or semi-minor axis of the ellipse.

**Segment** is known as the focal axis of the ellipse.

If the Pythagorean Theorem is applied to the previous figure, the relation between * a*,

*and*

**b***will be as follows:*

**c**## Ordinary Form of the Ellipse Equation

Considering the figure, it is a horizontal ellipse with center at the origin * C(0,0)*.

The equation that represents this ellipse is:

This expression corresponds to the ordinary equation of the ellipse with center at the origin and focal axis parallel to the * x* axis.

### Characteristics:

- The graph is symmetric with respect to the
and**x**axes.**y** - The coordinates of the foci are
.**F(±c,0)** - The coordinates of the vertices are
.**V(±a,0)** - The coordinates of the endpoints of the minor axis are
.**B(0,±b)** - The midpoint of the focal axis is known as the center, which in this case, is the origin.
- The distance of the segment is the major axis.
- The distance of the segment is the minor axis, so
is satisfied.**a > b** - The distance of the segment is the focal axis.
- The eccentricity of the ellipse is defined as the relation between the axis
and the axis**2c**, that is: . If**2a**then**c = 0**and the ellipse will become a circle. If**e = 0**, then we have a straight line.**c = a** - The latus rectum of the ellipse is the perpendicular chord to the major axis that passes through each focus and is given by the relation: .
- The directrix of the ellipse is a straight line that is at a distance from the vertex on the opposite side where the focus is found and is defined with the following equation:

Considering the figure, it is a vertical ellipse with center at the origin * C(0,0)*.

The equation that represents this ellipse is:

This expression corresponds to the ordinary equation of the ellipse with center at the origin and focal axis parallel to the * y* axis.

### Characteristics:

- The graph is symmetric with respect to the
and**x**axes.**y** - The coordinates of the foci are
.**F(0, ±c)** - The coordinates of the vertices are
.**V(0, ±a)** - The coordinates of the endpoints of the minor axis are
.**B(±b, 0)** - The midpoint of the focal axis is known as the center, which in this case is the origin.
- The distance of the segment is the major axis.
- The distance of the segment is the minor axis, so
is satisfied.**a > b** - The distance of the segment is the focal axis.
- The eccentricity of the ellipse is defined as the relation between the axis
and the axis**2c**, that is: . If**2a**then**c = 0**and the ellipse will become a circle. If**e = 0**then we have a straight line.**c = a** - The latus rectum of the ellipse is the perpendicular chord to the major axis that passes through each focus and is given by the relation: .
- The directrix of the ellipse is a straight line that is at a distance from the vertex on the opposite side where the focus is found and is defined with the following equation:

#### Example: Ellipse with Center at the Origin

Determine the equation of the ellipse that has its center at the origin, eccentricity and one focus at (3,0).

##### Solution:

Since one focus is at (3,0), we know that the ellipse is horizontal and that * c = 3*. So, its equation is:

With the given eccentricity, we obtain:

From the equation we solve for * b*:

Substitute the values for * a* and

*in the equation:*

**b****Answer:** The resulting equation is:

#### Example: Ellipse with Center at the Origin and Given Latus Rectum

Find the equation of the ellipse with center at the origin, one vertex at V(0,6) and LR = .

##### Solution:

According to the coordinates of the vertex and the characteristics of the ellipse, it is vertical and * a = 6*. So, the following equation is used:

With the latus rectum and * a = 6*, we obtain:

Substitute the values for * a = 6* and

*:*

**b = 4****Answer:** The resulting equation is:

### Ellipse with Center at (h, k)

Considering the figure, it is a horizontal ellipse with center at * C(h,k)*.

The equation that represents this ellipse is:

#### Elements:

**Eccentricity:**

**Center: (h,k)**

**Latus Rectum:**

**Vertices: V(h±a,k)**

**Directrix lines:**

**Foci: F(h±c,k)**

**Endpoints of minor axis: B(h,k±b)**

Considering the figure, it is a vertical ellipse with center at * C(h,k)*.

The equation that represents this ellipse is:

#### Elements:

**Eccentricity:**

**Center: (h,k)**

**Latus Rectum:**

**Vertices: V(h,k±a)**

**Directrix lines:**

**Foci: F(h,k±c)**

**Endpoints of minor axis: B(h±b,k)**

#### Example: Identifying Elements of an Ellipse

From the following equation of the ellipse, identify all of its elements and determine if it is horizontal or vertical:

##### Solution:

The condition * a > b* must be satisfied. Comparing the given equation with the standard forms, it belongs to:

Therefore, it is a vertical ellipse.

Comparing both equations, we obtain the following information:

By knowing the values for * a*,

*and*

**b***, we can obtain the missing information:*

**c****Answer:**

- The ellipse is vertical.
- Center (-5,-2).
- Vertices: V(-5,3) and V’(-5,-7).
- Foci: F(-5,2) and F’(-5,-6).
- Endpoints of minor axis: B(-2,-2) and B’(-8,-2).
- Eccentricity:
- Latus rectum:
- Directrix lines: ; ,

#### Example: Finding the Equation of an Ellipse

Find the equation of the ellipse with center at C(4,-2), a vertex at V(10,-2) and a focus at F(0,-2).

##### Solution:

The center, vertex, and focus have the same * y* value, so the focal axis is parallel to the

*axis. That is, the ellipse is horizontal, so the following formula is used:*

**x**Since we have the center C(h,k)=C(4,-2):

The distance between the center and the vertex is the value for * a*, and the distance between the center and the focus is the value for

*:*

**c**The distance between C(4,-2) and V(10,-2) is 6 units => * a = 6*. The distance between C(4,-2) and F(0,-2) is 4 units =>

*.*

**c = 4**From the relation we can obtain the value for * b*:

Substitute the values for * a* and

*in the equation:*

**b****Answer:** The resulting equation is: