Standard Form of Ellipse Equation with Formula and Solved Examples
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 an Ellipse Explained with Standard Forms
1. What is the equation of an ellipse?
The standard equation of an ellipse centered at the origin is x²/a² + y²/b² = 1. Here, a is the semi-major axis and b is the semi-minor axis. If a > b, the major axis is along the x-axis; if b > a, it is along the y-axis. For an ellipse centered at (h, k), the equation becomes (x − h)²/a² + (y − k)²/b² = 1.
2. What is the formula for the foci of an ellipse?
The foci of an ellipse are located at (±c, 0) or (0, ±c), where c = √(a² − b²). For a horizontal ellipse (a > b), the foci are (±c, 0). For a vertical ellipse (b > a), the foci are (0, ±c). The value c measures the distance from the center to each focus.
3. How do you find the center of an ellipse from its equation?
The center of an ellipse in standard form (x − h)²/a² + (y − k)²/b² = 1 is (h, k). To find it, rewrite the equation by completing the square if necessary. The constants inside the brackets give the center coordinates directly. For example, in (x − 3)²/9 + (y + 2)²/4 = 1, the center is (3, −2).
4. How do you find the major and minor axes of an ellipse?
The major and minor axes are determined by comparing the denominators in x²/a² + y²/b² = 1. The larger denominator corresponds to the major axis, and its length is 2a. The smaller denominator corresponds to the minor axis, and its length is 2b. For example, in x²/16 + y²/9 = 1, the major axis length is 8 and the minor axis length is 6.
5. What is the eccentricity of an ellipse?
The eccentricity of an ellipse is e = c/a, where c = √(a² − b²). It measures how stretched the ellipse is, with 0 < e < 1. If e is close to 0, the ellipse is nearly circular; if e is close to 1, it is more elongated. For example, if a = 5 and b = 4, then c = 3 and e = 3/5.
6. How do you graph an ellipse step by step?
To graph an ellipse, first identify its center, axes, and vertices from the standard equation. Follow these steps:
- Write the equation in standard form.
- Locate the center (h, k).
- Determine a and b from the denominators.
- Plot the vertices at (h ± a, k) or (h, k ± a).
- Plot the co-vertices at (h ± b, k) or (h, k ± b).
- Sketch a smooth curve through these points.
This method works for both horizontal and vertical ellipses.
7. What is the difference between a circle and an ellipse?
A circle is a special case of an ellipse where a = b. In a circle, the equation is x²/r² + y²/r² = 1, which simplifies to x² + y² = r². An ellipse has two unequal axes (a ≠ b), while a circle has equal radii in all directions. Also, the eccentricity of a circle is 0, whereas for an ellipse it lies between 0 and 1.
8. How do you find the equation of an ellipse given its vertices and co-vertices?
To find the equation, first determine the center and lengths of the semi-axes from the given vertices and co-vertices. Steps:
- The midpoint of the vertices gives the center (h, k).
- The distance from the center to a vertex gives a.
- The distance from the center to a co-vertex gives b.
- Substitute into (x − h)²/a² + (y − k)²/b² = 1.
For example, if vertices are (±5, 0) and co-vertices are (0, ±3), the equation is x²/25 + y²/9 = 1.
9. What is the general form of the equation of an ellipse?
The general form of an ellipse is Ax² + By² + Cx + Dy + E = 0, where A and B have the same sign and A ≠ B. To convert it into standard form, group x and y terms and complete the square. This process reveals the center, semi-major axis, and semi-minor axis.
10. What are the vertices and co-vertices of an ellipse?
The vertices are the endpoints of the major axis, and the co-vertices are the endpoints of the minor axis. For the ellipse x²/a² + y²/b² = 1 with a > b:
- Vertices: (±a, 0)
- Co-vertices: (0, ±b)
If the major axis is vertical, these positions switch accordingly. These points help in graphing and understanding the geometry of the ellipse.
