Ellipse Formula

Ellipse Formula

An ellipse is a geometric shape defined as the set of all points on a plane whose distances from two fixed points (called foci) are equal. It is commonly referred to as a stretched or elongated circle because it preserves the same curved shape but has two differing radii, known as the major and minor axes. Learn more about Ellipse and its formulas and solved examples based on it in this article

What is an Ellipse?

An ellipse is a closed-shape structure on a two-dimensional plane. It therefore, covers space in a 2D Plane. In Mathematics, an ellipse is the location of points in a plane so that their separation from a fixed point has a set ratio of “e” to their separation from a fixed line (less than 1).

Equation of Ellipse

The standard form equation of an ellipse centered at the origin with major axis parallel to the x-axis is:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

where:

• \( a \) is the length of the semi-major axis (half of the major axis)
• \( b \) is the length of the semi-minor axis (half of the minor axis)

Standard Equation of Ellipse

The standard equation of an ellipse is a mathematical representation that describes the shape and size of an ellipse in a coordinate plane. It is typically expressed in terms of \(x\) and \(y\) coordinates and can be written in two different forms, depending on the orientation of the major and minor axes of the ellipse.

Standard Equation of an Ellipse with Center at the Origin

The standard equation of an ellipse with its center at the origin (0,0) and major axis parallel to the x-axis is:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

where:

• \(a\) is the length of the semi-major axis (half of the major axis).
• \(b\) is the length of the semi-minor axis (half of the minor axis).

Standard Equation of an Ellipse with Center at (h, k)

If the center of the ellipse is at the point (h, k), then the standard equation becomes:

\[ \frac{(x – h)^2}{a^2} + \frac{(y – k)^2}{b^2} = 1 \]

where:

• \( (h, k) \) is the center of the ellipse.
• \(a\) and \(b\) are the same as in the previous equation

Derivation of Ellipse Equation

Start with the definition of an ellipse as the set of points where the sum of the distances from two fixed points (foci) is constant. Let the foci be \( F_1 \) and \( F_2 \), and let this constant sum be \( 2a \).

Consider a point \( P(x, y) \) on the ellipse. By the distance formula, the distance from \( P \) to each focus is given by \( d_1 = \sqrt{(x – x_1)^2 + (y – y_1)^2} \) and \( d_2 = \sqrt{(x – x_2)^2 + (y – y_2)^2} \).

 According to the definition of an ellipse, \( d_1 + d_2 = 2a \).

Square both sides of the equation to eliminate the square roots: \( (d_1 + d_2)^2 = (2a)^2 \).

Substitute the expressions for \( d_1 \) and \( d_2 \) into the equation: \( \left(\sqrt{(x – x_1)^2 + (y – y_1)^2} + \sqrt{(x – x_2)^2 + (y – y_2)^2}\right)^2 = (2a)^2 \).

Expand the left side of the equation using the binomial theorem.

Simplify the equation by canceling out like terms and combining similar terms.

This simplification leads to the general equation of an ellipse in standard form: \( \frac{(x – h)^2}{a^2} + \frac{(y – k)^2}{b^2} = 1 \), where \( (h, k) \) is the center of the ellipse, \( a \) is the length of the semi-major axis, and \( b \) is the length of the semi-minor axis.

Properties of Ellipse

Ellipses possess several important properties:

1. Foci : An ellipse has two foci, denoted as \(F_1\) and \(F_2\). The sum of the distances from any point on the ellipse to the two foci is constant.
2. Major and Minor Axes : The major axis is the longest diameter of the ellipse, passing through both foci and through the center. The minor axis is perpendicular to the major axis and passes through the center.
3. Vertices : The endpoints of the major axis are called the vertices. They lie on the ellipse’s perimeter and are equidistant from both foci.
4. Center : The center of the ellipse is the midpoint of the major axis, equidistant from the two foci.
5. Eccentricity : The eccentricity (\(e\)) of an ellipse quantifies how “stretched out” it is. It is defined as the ratio of the distance between the center and one of the foci to the length of the semi-major axis (\(a\)).
6. Latus Rectum : The latus rectum is the chord passing through a focus and perpendicular to the major axis. Its length is given by \( \frac{2b^2}{a} \), where \(a\) is the length of the semi-major axis and \(b\) is the length of the semi-minor axis.

Ellipse Formulas

1. Standard Form Equation of an Ellipse :

\[ \frac{(x – h)^2}{a^2} + \frac{(y – k)^2}{b^2} = 1 \]

Where:

• \( (h, k) \) is the center of the ellipse,
• \( a \) is the length of the semi-major axis,
• \( b \) is the length of the semi-minor axis.

2. Eccentricity : \[ e = \sqrt{1 – \frac{b^2}{a^2}} \]

Where \( e \) is the eccentricity of the ellipse.

3. Length of Major Axis : \[ 2a \]
4. Length of Minor Axis : \[ 2b \]
5. Foci Coordinates : Foci lie on the major axis. The distance from the center to each focus is \( c \), where \( c^2 = a^2 – b^2 \). Foci coordinates: \( (h \pm c, k) \).
6. Vertices Coordinates : Vertices lie on the major axis. Vertices coordinates: \( (h \pm a, k) \).
7. End Points of Minor Axis : End points of the minor axis lie on the ellipse’s perimeter. End points coordinates: \( (h, k \pm b) \).
8. Area of Ellipse : \[ A = \pi ab \]
9. Perimeter of Ellipse (Approximation) :

There’s no simple closed-form expression for the perimeter of an ellipse. However, you can use various approximations such as Ramanujan’s formula or numerical integration. P ≈ π [ (3/2)(a+b) – √(ab) ]

10. Latus Rectum of Ellipse

The length of the latus rectum of an ellipse depends on its semi-major axis (\(a\)) and semi-minor axis (\(b\)). The formula for the length of the latus rectum (\(l\)) is: \[ l = \frac{2b^2}{a} \]

This formula represents the distance between the two points on the ellipse’s major axis through which the directrices (lines that are equidistant from the foci) pass.

How to Draw Ellipse

1. Determine the Center:

Identify the coordinates of the center of the ellipse. Denote them as \( (h, k) \).

2. Determine the Axes Lengths:

Determine the lengths of the semi major axis (\(a\)) and semi minor axis (\(b\)) of the ellipse. These represent half the lengths of the major and minor axes, respectively.

3. Plot the Vertices:

Use the center \( (h, k) \) as the starting point. Plot the vertices of the ellipse. The vertices are located at a distance of \(a\) units from the center along the major axis in both directions.

4. Plot the Co vertices:

Plot the co vertices of the ellipse. The co vertices are located at a distance of \(b\) units from the center along the minor axis in both directions.

5. Connect the Points:

Connect the vertices and co vertices using a smooth curve. Remember that an ellipse is a conic section, so it should have a smooth, continuous shape.

6. Adjust for Eccentricity (if necessary):

If the ellipse is eccentric (not a circle), the distance between the center and each focus will be different. Adjust the curve accordingly to ensure that the sum of the distances from any point on the ellipse to the two foci remains constant.

7. Check the Eccentricity (optional):

Calculate the eccentricity of the ellipse using the formula \(e = \sqrt{1 \frac{b^2}{a^2}}\). This will give you a measure of how stretched out or squished the ellipse is.

Graphing Ellipse

Graphing an ellipse involves plotting points based on its standard equation and then drawing the curve. Here are the steps for graphing an ellipse given its standard form equation

1. Standard Form of Ellipse Equation

For a horizontal ellipse \[ \frac{(x h)^2}{a^2} + \frac{(y k)^2}{b^2} = 1 \]

For a vertical ellipse \[ \frac{(x h)^2}{b^2} + \frac{(y k)^2}{a^2} = 1 \]

Where \( (h, k) \) is the center, \( a \) is the semi major axis, and \( b \) is the semi minor axis.

2. Identify the Center

The center of the ellipse is at \( (h, k) \).

3. Determine the Axes Lengths

Identify \( a \) and \( b \). \( a \) is the distance from the center to the vertices along the major axis, and \( b \) is the distance from the center to the co vertices along the minor axis.

4. Plot the Center

Plot the center of the ellipse on the coordinate plane at \( (h, k) \).

5. Plot the Vertices

For a horizontal ellipse Vertices are at \( (h \pm a, k) \).

For a vertical ellipse Vertices are at \( (h, k \pm a) \).

6. Plot the Co vertices

For a horizontal ellipse Co vertices are at \( (h, k \pm b) \).

For a vertical ellipse Co vertices are at \( (h \pm b, k) \).

7. Draw the Ellipse

Draw a smooth curve passing through the vertices and co vertices, forming an oval shape.

8. Optional Plot Additional Points

To ensure accuracy, you can plot additional points by choosing \( x \) values within the range \([h a, h + a] and solving for \( y \) using the ellipse equation, or vice versa for \( y \) values.

Solved Examples on Ellipse Formula

Example 1: Determine the foci of the ellipse given by the equation \(\frac{x^2}{16} + \frac{y^2}{9} = 1\).

Solution :

Identify \(a\) and \(b\) :

\[ a^2 = 16 \implies a = 4 \]

\[ b^2 = 9 \implies b = 3 \]

Find the focal distance (c) :

\[ c^2 = a^2 – b^2 \]

\[ c^2 = 16 – 9 \]

\[ c^2 = 7 \]

\[ c = \sqrt{7} \]

Foci positions :

Since the major axis is along the x-axis, the foci are located at \((\pm c, 0)\).

So, the foci are at:

\[ (\sqrt{7}, 0) \quad \text{and} \quad (-\sqrt{7}, 0) \]

Example 2: Find the area of an ellipse with a semi-major axis of length 8 and a semi-minor axis of length 6.

Solution :

Identify \(a\) and \(b\) :

\[ a = 8 \]

\[ b = 6 \]

Use the area formula :

\[ A = \pi a b \]

Calculate the area :

\[ A = \pi \times 8 \times 6 \]

\[ A = 48\pi \]

So, the area of the ellipse is:

\[ 48\pi \text{ square units} \]

Example 3: If the semi-major axis is 10 cm long and the semi-minor axis is an ellipse with a 7 cm diameter. Locate the location.

Solution:

Given that a = 10 cm for an ellipse’s semi-major axis

b = 7 cm is the length of an ellipse’s semi-minor axis.

Using the formula, we can get an ellipse’s area;

Area = πab

= π x 10 x 7

= 70 x π

Area is thus equal to 219.91 cm².