An ellipse is a set of points such that the sum of the distances from any point on the ellipse to two fixed points (foci) is constant. In this article, we will learn about the ellipse definition, Ellipse formulas, and others in detail.
What is Ellipse?
An ellipse is a set of points in a plane, the sum of whose distances from two fixed points in the plane is constant. The two fixed points are called foci of the ellipse. The figure below shows the two fixed points and shows how an ellipse can be traced from those points.
Note: Constant distance mentioned above should always be less than the distance between the two focal points.
The figure below shows a labelled diagram of the ellipse.
Ellipse DefinitionThe line segment joining and going through two foci is called the major axis. The mid-point of the line segment between two foci is called the centre of the ellipse. The line perpendicular to the major axis and passing through the centre of the ellipse is called the minor axis. The endpoints of the major axis which cut the ellipse are called vertices of the ellipse.
Let's say the length of the major axis is 2a and while that of the minor axis is 2b. The distance between two foci is defined as 2c.
Standard form of the equation of an ellipse centered at the origin is:
x2/a2 + y2/b2 = 1
If the ellipse is centered at (h, k), the equation becomes:
(x - h)2/a2 + (y - k)2/b2 = 1
where:
- a is the length of the semi-major axis (horizontal radius)
- b is the length of the semi-minor axis (vertical radius)
Now the ellipse formula is,
Take a point P at one end of the major axis. Total of the distances between point P and the foci is,
F1P + F2P = F1O + OP + F2P = c + a + (a–c) = 2a
Then, select a point Q on one end of the minor axis. The sum of the distances between Q and the foci is now,
F1Q + F2Q = √ (b2 + c2) + √ (b2 + c2) = 2√ (b2 + c2)
We already know that points P and Q are on the ellipse. As a result, by definition, we have
2√ (b2 + c2) = 2a
then √ (b2 + c2) = a
i.e. a2 = b2 + c2 or c2 = a2 – b2
The following is the equation for ellipse.
c2= a2 – b2
We have defined the major and minor axes in the previous sections. Now the question that comes to mind is that are they related to each other in some way? Are there any special cases of the ellipse? First, let's establish the relationship between the major and the minor axes.
Consider point A. The distance of A from the two Centre is,
AF1 + AF2 = OF1 + OA + AF2
= c + a + a - c = 2a
From point B on the minor axis.
F1B + F2B = 2\sqrt{b^2 + c^2}
We know from the definition of the ellipse that,
AF1 + AF2 = F1B + F2B
2a = 2\sqrt{b^2 + c^2}\\ a = \sqrt{b^2 + c^2}
The eccentricity of an ellipse is the ratio of the distances from the Centre of the ellipse to one of the foci and to one of the vertices of the ellipse
e = c/a
We know that,
c2 = a2 – b2
e = √[1-(b2/a2)]
Latus Rectum are line segments perpendicular to the major axis of the ellipse and passing through any of the foci of the ellipse in such a manner that their endpoints always lie on the ellipse.
Length of the latus rectum for the ellipse is given by the formula,
L = 2b2/a
where,
- a is the minor axis
- b is the major axis
Area of an ellipse is given by the formula:
Area of Ellipse = π × (Semi-Major Axis) × (Semi-Minor Axis)
Area of Ellipse = πab
where,
- a is the minor axis
- b is the major axis
Perimeter of an ellipse is given by the formula:
P = 2π√{( a2 + b2) / 2}
where,
- a is the minor axis
- b is the major axis
Article Related to Ellipse Formula:
Example 1: Find the equation of ellipse if the endpoints of the major axis lie on (-10,0) and (10,0) and endpoints of the minor axis lie on (0,-5) and (0,5).
Solution:
Since the major axis is x-axis, the ellipse equation should be,
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
2a = 20
⇒a = 10
2b = 10
⇒b = 5
\frac{x^2}{10^2} + \frac{y^2}{5^2} = 1
Example 2: Find the equation of an ellipse with origin as centre and x-axis as major axis. Given that the distance between two foci is 10cm, e = 0.4 and b = 4cm
Solution:
Standard equation of the ellipse is,
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
We know b = 4, e = 0.4 and c = 10.
c = ae \\ 10 = a(0.4) \\ 25 = a
Thus, now we have a = 25 and b = 4
So, the equation of ellipse is,
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \\ \frac{x^2}{25^2} + \frac{y^2}{4^2} = 1
Example 3: Find the equation of an ellipse whose major axis is 40cm and foci lie on (5,0) and (-5,0).
Solution:
\frac{x^2}{a^2} + \frac{x^2}{b^2} = 1
a = \frac{40}{2} = 20
We know c = 10
c2 = a2 - b2
102 = 202 - b2
b2 = 202 - 102
b2 = 300
Thus, the equation becomes,
\frac{x^2}{400} + \frac{x^2}{300} = 1
Example 4: Find the equation of an ellipse whose major axis is 40cm and foci lie on (0,5) and (0,-5).
Solution:
Since the foci lie on y-axis. The major axis is on y-axis. Thus, the ellipse is of the form,
\frac{x^2}{b^2} + \frac{x^2}{a^2} = 1
a = \frac{40}{2} = 20
We know c = 10
c2 = a2 - b2
102 = 202 - b2
b2 = 202 - 102
b2 = 300
Thus, the equation becomes,
\frac{x^2}{300} + \frac{x^2}{400} = 1
Example 5: Find the equation of ellipse if the major axis is the x-axis and the minor axis is the y-axis and (4,3) and (-1,4) lie on the ellipse.
Solution:
Standard equation of the ellipse is,
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
These points must satisfy the equation. (4,3) and (-1, 4).
\frac{4^2}{a^2} + \frac{(-3)^2}{b^2} = 1 \\ \frac{16}{a^2} + \frac{9}{b^2} = 1 \\
\frac{(-1)^2}{a^2} + \frac{4^2}{b^2} = 1 \\ \frac{1}{a^2} + \frac{16}{b^2} = 1 \\
Let's say, x = \frac{1}{a} \text{ and } y = \frac{1}{b}
16x + 9b = 1
x + 16b = 1
Solving the equations,
We find that a = \frac{247}{7} \text{ and } b = \frac{247}{15}
- Find the equation for an ellipse with a major axis length of 10 and a minor axis length of 6 units.
- Calculate the eccentricity of an ellipse with a = 5 and b = 3.
- Calculate the foci coordinates for the ellipse x2/16+y2/9=1.
- What is the latus rectum length for an ellipse whose equation is x2/25+y2/9=1?
- Determine the lengths of an ellipse's axes given its equation, 9x2+4y2=36.
- Convert the general equation 25x2+9y2−150x+54y−36 = 0. Fill out the standard form and identify the key parameters.
- Calculate the area of the ellipse x2/4+y2/1=1.
- Find the equation for an ellipse with foci at (0, ±5) and a minor axis length of 6 units.
- Calculate the length of the major axis if the ellipse has an eccentricity of 0.8 and a minor axis length of 4. A satellite orbits the Earth in an elliptical path, with the Earth as its focal point. If the orbit's semi-major axis is 20,000 km and the eccentricity is 0.5, calculate the distance from Earth to the orbit's nearest point.
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