What is an Ellipse?
Ellipses for the first time was discovered by ancient greeks when they were studying the conic section. Ellipse was achieved when the right corner of it was sliced at different angles. Given below is a demonstration of how the right angle of an ellipse can be sliced at different angles.
The resultant of the intersection of a right circular cone with a plane is called a conic section, or conic. Conic is basically a shape that is determined by the angle at which the plane intersects the cone. It can also be described by a set of points in the coordinate plane which can also be represented by the graph of any quadratic equation in two variables depending on the signs of the equations and the coefficients of the variable.
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More About Ellipse
An ellipse and a circle look almost the same, the only difference is that an ellipse is slightly squashed into an oval. You can say it’s like a line bending around until the two of its ends meet, just like a circle. Things having a shape like an ellipse is known as 'elliptical'.
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Focus: Ellipse is denoted by a set of points(x,y) in a plane such that the sum of their distances from two fixed points is always constant. These fixed points are called foci (or focus if singular) of the ellipse.
Major Axis And Minor Axis: Every ellipse has two axes of symmetry called the major axis and minor axis. The longer axis known as the major axis has its endpoint as the vertex of the ellipse. Similarly, the shorter axis known as the minor axis has its endpoint as a co-vertex of the ellipse. The foci always rest on the major axis, and the sum of the distances between the foci and any other point on the ellipse (the constant sum) is always greater than the distance between the foci.
Centre of an Ellipse: The centre of an ellipse is the common point and also the midpoint of both the major and minor axes. Both the axes (major and minor) are perpendicular to the centre.
The General Equation of Ellipse
There is a standard form of the general equation of ellipse.
\[\frac{x^2}{a^2}\] + \[\frac{y^2}{b^2}\] = 1
Ellipses are usually positioned in two ways - vertically and horizontally. Ellipses are said to be vertical in the coordinate plane if the axes on x– and y-axes whereas it is said to be horizontal in the coordinate plane if the axes lie parallel to the x– and y-axes. Apart from these, ellipses can also be rotated in the coordinate plane.
There are two cases to work with horizontal and vertical ellipses in the coordinate plane:
i) Ellipses that are centred at the origin and
ii) Ellipses that are centred at a point other than the origin.
The general equation of ellipses in a standard form or say standard equation of ellipse is given below:
\[\frac{x^2}{a^2}\] + \[\frac{y^2}{b^2}\]
Derivation of Equations of Ellipse
When the centre of the ellipse is at the origin and the foci are on the x-axis or y-axis, then the standard equation of ellipse can be derived as shown below.
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Now its time for deriving the standard equation of ellipse that is shown in Fig.5 (a) with the foci on the x-axis. Let F1 and F2 be the foci and O be the mid-point of the line F1F2. Also, let O be the origin and the line from O through F2 be the positive x-axis and that through F1 as the negative x-axis.
Further, let the line drawn through O perpendicular to the x-axis be the y-axis. Let the coordinates of F1 be (– c, 0) and F2 be (c, 0) as shown in Fig.5 (a) above.
Now, we take a point P(x, y) on the ellipse such that, PF1 + PF2 = 2a
By the distance formula, we have,
√{(x + c) 2 + y2} + √{(x – c) 2 + y2} = 2a
Or, √ {(x + c)2 + y2} = 2a – √ {(x – c)2 + y2}
Further, let’s square both sides. Hence, we have
(x + c) 2 + y2 = 4a2 – 4a√{(x – c) 2 + y2} + (x – c) 2 + y2
Simplifying the equation, we get √ {(x – c)2 + y2} = a – x(c/a)
Now we can square both the sides again and just simplify it further to get,
x2/a2 + y2/(a2 – c2) = 1
We know that c2 = a2 – b2. Therefore, we have x2/a2 + y2/b2 = 1
Therefore, we can say that any point on the ellipse satisfies the equation:
x2/a2 + y2/b2 = 1 …
Solved Examples
Example 1) Use the formula to find the coordinates of foci when the major axis is 5 and the minor axis is 3.
Solution 1) Given the formula F = \[\sqrt{j^2-n^2}\]
F = \[\sqrt{5^2-3^2}\]
F = \[\sqrt{25-9}\]
F = \[\sqrt{16}\]
F = 4
Foci = (0,4) & (0,-4)
Example 2) Use the formula to find the coordinates of foci when the major axis is 10 and the minor axis is 6.
Solution 2) Given the formula F = \[\sqrt{j^2-n^2}\]
F = \[\sqrt{10^2-6^2}\]
F = \[\sqrt{100-36}\]
F = \[\sqrt{64}\]
F = 8
Foci = (0,8) & (0,-8)
FAQs on Equations Of Ellipse
1. What are the standard equations of an ellipse as per the CBSE Class 11 syllabus?
According to the CBSE syllabus for the 2025-26 session, there are two standard forms for the equation of an ellipse centred at the origin (0,0):
- Horizontal Ellipse: The equation is x²⁄a² + y²⁄b² = 1. Here, the major axis is along the x-axis, and a > b > 0. The value 'a' represents the semi-major axis, and 'b' represents the semi-minor axis.
- Vertical Ellipse: The equation is x²⁄b² + y²⁄a² = 1. In this case, the major axis is along the y-axis, and again, a > b > 0.
2. What does the eccentricity of an ellipse represent?
The eccentricity (e) of an ellipse is a measure of how much it deviates from being a perfect circle. It is calculated by the formula e = c/a, where 'c' is the distance from the centre to a focus, and 'a' is the distance from the centre to a vertex. The value of eccentricity for an ellipse is always between 0 and 1 (i.e., 0 ≤ e < 1). A value of e=0 indicates a circle, while a value approaching 1 indicates a very elongated or flat ellipse.
3. What is the key difference in the equations of a horizontal and a vertical ellipse?
The key difference lies in the denominators of the x² and y² terms. In the standard form, 'a' always represents the semi-major axis and is always greater than 'b' (the semi-minor axis).
- For a horizontal ellipse, the larger denominator (a²) is under the x² term, indicating the major axis lies along the x-axis.
- For a vertical ellipse, the larger denominator (a²) is under the y² term, indicating the major axis lies along the y-axis.
4. How is an ellipse fundamentally different from a circle?
A circle is actually a special case of an ellipse. The fundamental difference lies in their defining properties related to foci. A circle is defined by a single center point, with all points on its circumference being equidistant (the radius) from it. An ellipse, however, is defined by two fixed points called foci. An ellipse is the set of all points in a plane where the sum of the distances from the two foci is constant. When the two foci of an ellipse merge into a single point, it becomes a circle.
5. What are the formulas for the area and perimeter of an ellipse?
The formulas for the area and perimeter of an ellipse are quite different in complexity:
- Area: The formula is simple and exact: Area = πab, where 'a' is the semi-major axis and 'b' is the semi-minor axis.
- Perimeter (Circumference): There is no simple exact formula for the perimeter of an ellipse. It is calculated using complex integrals. However, a widely used and accurate approximation is Ramanujan's formula: P ≈ π[3(a+b) - √((3a+b)(a+3b))].
6. How do you convert the general equation of an ellipse to its standard form?
To convert the general second-degree equation into the standard form of an ellipse, you use the method of completing the square. The key steps are:
- Group the x-terms and y-terms together and move the constant term to the other side.
- Factor out the coefficients of x² and y² from their respective groups.
- Complete the square for both the x-variable and y-variable expressions, remembering to add the same values to the other side of the equation.
- Rewrite the completed squares as squared binomials, for example, (x-h)² and (y-k)².
- Divide the entire equation by the constant on the right-hand side to make it equal to 1. The resulting equation will be in the standard form (x-h)²⁄a² + (y-k)²⁄b² = 1.
7. Where can we see examples of ellipses in real life?
Ellipses are very common in the world around us. Some notable real-world examples include:
- The orbits of planets, comets, and satellites. For instance, Earth follows an elliptical path around the Sun, with the Sun at one of the foci.
- The design of whispering galleries, where a sound whispered at one focus can be heard clearly at the other focus.
- The cross-sectional shape of many objects, like an egg or a rugby ball.
- Elliptical arches in bridges and architecture, which are known for their strength and aesthetic appeal.
