Equation of Ellipse – Definition, Parametric Form, Properties & Examples

An ellipse is a special shape in mathematics that comes under the topic of conic sections. It looks like a stretched-out circle or an oval. Just like a circle, an ellipse is made by points, but instead of being the same distance from the center (like in a circle), the points in an ellipse follow a different rule — the total distance from any point on the ellipse to two fixed points (called foci) stays the same.

An ellipse is different from a circle because it's not perfectly round. A circle has all points equally far from the center, but in an ellipse, some points are farther than others.

Ellipse definition: An ellipse is a special shape that looks like a stretched circle. It is formed by all the points on a flat surface such that the total distance from any point on the shape to two fixed points (called foci) is always the same. These two fixed points are located inside the ellipse, and every point on the ellipse keeps the sum of its distances to both foci constant. This unique rule gives the ellipse its oval shape. You can think of it like tying a string between two pins and tracing the curve with a pencil while keeping the string tight.

An ellipse can also be defined as the locus of a point that travels in a plane such that the ratio of its distance from an established point (focus) to a fixed straight position (directrix) is constant and less than unity i.e. eccentricity e < 1. Eccentricity is a factor of the ellipse, which demonstrates its elongation and is denoted by ‘e’.

Parts of an Ellipse

• Foci (plural of focus): An ellipse has two special points inside it called foci. Their positions are usually written as F(c, 0) and F’(-c, 0). These points help define the shape of the ellipse.
• Centre: The centre of the ellipse is the point exactly in the middle of the two foci. It's the midpoint of the line that connects the foci.
• Latus Rectum: This is a line that is drawn perpendicular (at a right angle) to the main axis of the ellipse and passes through each focus. It helps describe the shape more clearly.
• Transverse Axis: This is the main horizontal line of the ellipse that goes through the centre and both foci.
• Conjugate Axis: This is the vertical line that also goes through the centre, but it is perpendicular to the transverse axis.
• Eccentricity (e): This is a number that tells us how stretched the ellipse is. It is found by dividing the distance from the centre to a focus by the distance from the centre to the end of the ellipse. Its value is always less than 1 for ellipses.
• Principal Axis: This is just another name for the line that joins both foci. Its midpoint is also the centre of the ellipse.

The major axis of an ellipse is the longest line that passes through the center of the ellipse and touches both ends. It connects the two farthest points on the boundary of the ellipse and goes right through the middle. This is why it’s also called the longest diameter of the ellipse.

The minor axis is the shortest line that also passes through the center of the ellipse but connects the two nearest points on the edge. It’s like the shortest path across the ellipse through the center, so it's called the shortest diameter.

We also have two more terms:

• The semi-major axis is just half the length of the major axis.
• The semi-minor axis is half the length of the minor axis.

Perimeter of Ellipse: The perimeter of an ellipse is the entire length run by its outer boundary. Formula to determine the perimeter of an ellipse is or ᠎ where a is the length of the semi-major axis and b is the length of the semi-minor axis.

Area of Ellipse: The area of an ellipse is the measure of the region present inside it. Alternatively, one can understand that the area of an ellipse is the total number of unit squares that can fit in it.

where “a” is the length of the semi-major axis and “b” is the length of the semi-minor axis.

The association between the semi-axes of the ellipse is represented by the following formula:

Eccentricity of Ellipse: The eccentricity of an ellipse is presented as the ratio of the length of the focus from the center of the ellipse, and the length of one end of the ellipse from the center of the ellipse.

Eccentricity of the ellipse is given as where “c” is the focal length and “a” denotes the length of the semi-major axis.

Latus Rectum of Ellipse: Latus rectum of an ellipse can be interpreted as the line drawn perpendicular to the transverse axis of the ellipse and crossing through the foci of the ellipse. The formula to obtain the length of the latus rectum of an ellipse can be addressed as:

Length of Latus Rectum= where “a” is the length of the semi-major axis and “b” is the length of the semi-minor axis.

There are 2 standard equations of ellipse. These equations of the ellipse are based on the transverse axis and the conjugate axis of the ellipse. The below image displays the two standard forms of equations of an ellipse.

Standard equations of ellipse are also known as the general equation of ellipse.

Standard equations of ellipse when ellipse is centered at origin with its major axis on X-axis:

In this form both the foci rest on the X-axis.

Standard equations of ellipse when ellipse is centered at origin with its major axis on Y-axis:

In this form both the foci rest on the Y-axis.

The equation of an ellipse formula assists in expressing an ellipse in the algebraic form. The formula to determine the equation of an ellipse can be given as:

Equation of the ellipse with center at (0,0) :

Equation of the ellipse with center at (h,k):

The standard form of the equation of an ellipse with center (h, k) and major axis parallel to the x-axis is given as

The standard form of the equation of an ellipse with center (h,k)and major axis parallel to the y-axis is given as

Equation of Ellipse in Parametric Form

Parametric equation of ellipse: is given by , and the parametric coordinates of the points lying on it are furnished by

Equation of Tangents and Normals to Ellipse

Equation of a tangent to the ellipse in Point form:

At the point equation of a tangent is presented by:

Equation of Tangent to Ellipse in Slope Form

Equation of tangent to ellipse in terms of slope m:

The slope is m and the coordinates of the point of contact are:

Equation of Tangent to Ellipse in Parametric form

Equation of normal to the ellipse

Learn about Parabola Ellipse and Hyperbola

Equation of Tangent to Ellipse in Point form

At the point the equation of normal to the ellipse is presented by:

OR

Equation of Normal to Ellipse Slope Form

Equation of normal to the ellipse in terms of slope m is presented by:

Equation of Normal to Ellipse Parametric form

Learn about Equation of Parabola 

How to Draw an Ellipse

Drawing an ellipse may seem tricky, but you can do it easily by following these steps:

1. Decide the Major Axis:
   First, choose how long you want the major axis to be. This is the longest line across the ellipse.

2. Draw the Major Axis:
   Draw a straight horizontal line using a ruler. This line represents the major axis.

3. Mark the Center:
   Find the middle point of the major axis. You can do this by measuring its length and dividing it by two.

4. Draw a Circle for Major Axis:
   Using a compass, draw a circle using the length of the major axis as the diameter.

5. Decide the Minor Axis:
   Now, choose the length of the minor axis. This is the shorter diameter of the ellipse.

6. Draw the Minor Axis:
   Place a protractor at the center of the major axis. Mark points at 90° and 270° to form a vertical line. This will be your minor axis.

7. Draw a Circle for Minor Axis:
   Draw another circle using the minor axis as the diameter, just like you did for the major axis.

8. Divide into Angles:
   Divide the circles into 12 equal parts (every 30 degrees) using the protractor. Mark these points clearly.

9. Draw Horizontal Lines:
   From each marked point on the inner (minor) circle, draw horizontal lines outward. These lines should be shorter near the center and longer near the ends.

10. Draw Vertical Lines:
    From each marked point on the outer (major) circle, draw vertical lines inward. These should be longer near the center and shorter near the ends.

11. Mark Intersections:
    The points where horizontal and vertical lines cross are points on the ellipse.

12. Connect the Points:
    Finally, use your hand or a curved ruler to smoothly connect all the crossing points. This will form the shape of the ellipse.

Solved Examples of Equation of Ellipse

Example 1: Determine the lengths of major and minor axes of the ellipse given by the equation:

Solution: The equation of the ellipse is:

The general equation of ellipse is:

On comparison:

Hence:

The length of the major axis = 2a =8.

The length of the minor axis = 2b = 6.

Example 2: The length of the semi-major and semi-minor axis of an ellipse is 4 cm and 2 cm respectively. Obtain its eccentricity and the length of the latus rectum.

Solution: To determine the eccentricity and the length of the latus rectum of an ellipse.

The general equation of ellipse is:

Given: a = 4 cm, and b = 2 cm.

The equation of the ellipse is:

Now, using ellipse formula for eccentricity:

Now, practicing ellipse formula for latus rectum:

Length of Latus Rectum=

Eccentricity and the length of the latus rectum of the ellipse are 0.866 cm and 2 cm respectively.

Check more topics of Mathematics here.

Example 3: If the length of the semi-major axis is 5cm and the semi-minor axis is 3cm of an ellipse. Find its area and perimeter.

Solution: Given, length of the semi-major axis of an ellipse, a = 5cm and the

length of the semi-minor axis of an ellipse, b = 3cm.

By the formula for perimeter of an ellipse:

Example 4: Find the lengths of major and minor axes of the ellipse

Solution: Given equation is:

On dividing both sides by 36 we get:

On comparing this ellipse equation with the standard one:

Form :᠎

In this form both the foci rest on the Y-axis.

For the above equation, the ellipse is centered at the origin with its major axis on the Y-axis.

We obtain

That is

The length of the major axis = 2a =6.

The length of the minor axis = 2b = 4.

Properties of an Ellipse

• Axes: An ellipse has two main lines called axes – the major axis and the minor axis. These two lines cross each other at a right angle (90°). The major axis is the longest straight line that goes across the ellipse, while the minor axis is the shortest.
• Eccentricity (e): This tells us how stretched or flat the ellipse looks. It's a number between 0 and 1. If e is closer to 0, the ellipse looks more like a circle. The formula is:
  e = distance between foci / length of major axis
• Center: The center is the exact point where the major and minor axes cross. It's like the midpoint of the ellipse.
• Vertices: These are the two end points of the major axis – the farthest points on the ellipse from the center along the longest line.
• Foci (plural of focus): These are two fixed points inside the ellipse. For every point on the ellipse, if you add the distance to both foci, that total stays the same. That’s what makes an ellipse unique!
• Latus Rectum: This is a small line that goes through one focus and is straight across, perpendicular to the major axis. It helps describe the shape of the ellipse. The length of the latus rectum is given by the formula:
  L = 2b²/a, where a is the semi-major axis and b is the semi-minor axis.

We hope that the above article on Equation of Ellipse is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.

If you are checking Equation of Ellipse article, also check the related maths articles:
Equation of hyperbola	Difference Between Parabola and Hyperbola
Equation of a Circle	Equation of Parabola
Equation of a Line	Quadratic Equation

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