Understanding the Ellipse: Equations, Characteristics, and Examples

Posted on Aug 23, 2024 in Physics

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

F and F’ are the foci of the ellipse.

B and B’ are the endpoints of the minor axis.

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, b and c will be as follows:

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 x and y axes.
• 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 a > b is satisfied.
• The distance of the segment is the focal axis.
• The eccentricity of the ellipse is defined as the relation between the axis 2c and the axis 2a, that is: . If c = 0 then e = 0 and the ellipse will become a circle. If c = a, then we have a straight line.
• 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 x and y axes.
• 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 a > b is satisfied.
• The distance of the segment is the focal axis.
• The eccentricity of the ellipse is defined as the relation between the axis 2c and the axis 2a, that is: . If c = 0 then e = 0 and the ellipse will become a circle. If c = a then we have a straight line.
• 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 b in the equation:

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, b and c, we can obtain the missing information:

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

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 b in the equation:

Answer: The resulting equation is: