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Foci of Ellipse

Foci of the ellipse, are the reference points in an ellipse and it helps to derive the equation of the ellipse. There are two foci for the ellipse. Also, the locus of the ellipse is defined as the sum of the distances from the two foci, as a constant value.

Let us learn more about the foci of the ellipse, its formulas, and solved examples.

1.	What Is Foci of Ellipse?
2.	How To Find Foci of Ellipse?
3.	Examples on Foci of Ellipse
4.	Practice Questions
5.	FAQs on Foci of Ellipse

What Is Foci of Ellipse?

The foci of the ellipse are the two reference points that help in drawing the ellipse. The foci of the ellipse lie on the major axis of the ellipse and are equidistant from the origin. An ellipse represents the locus of a point, the sum of the whose distance from the two fixed points are a constant value. These two fixed points are the foci of the ellipse. Let the point on the ellipse be P and the two fixed points be F and F' respectively. Here we have PF + PF' = C, a constant value.

For a standard form of an ellipse x²/a² + y²/b² = 1, the coordinates of the two foci are F (+ae, o), and F' (-ae, 0).

Foci of Ellipse

How to Find Foci of Ellipse?

The foci of the ellipse can be calculated by knowing the semi-major axis, semi-minor axis, and the eccentricity of the ellipse. The semi-major axis for an ellipse x²/a² + y²/b² = 1 is 'a', and the formula for eccentricity of the ellipse is e =\(\sqrt {1 - \frac{b^2}{a^2}}\). The abscissa of the coordinates of the foci is the product of 'a' and 'e'. The coordinates of the foci of the ellipse are (+ae, 0), and (-ae, 0) respectively.

For an ellipse (x - h)²/a² + (y - k)²/b² = 1, the center of the ellipse is (h, k), and the coordinates of foci are F (+(h + a)e, k), and F'((h - a)e, k).

Related Topics

The following topics would help in a better understanding of the foci of ellipse.

Examples on Foci of Ellipse

Example 1: Find the coordinates of the foci of ellipse having an equation x²/25 + y²/16 = 0.

Solution:

The given equation of the ellipse is x²/25 + y²/16 = 0.

Commparing this with the standard equation of the ellipse x²/a² + y²/b² = 1, we have a = 5, and b = 4

Let us first calculate the eccentricity of the ellipse.

e =\(\sqrt {1 - \frac{b^2}{a^2}}\) =\(\sqrt {1 - \frac{4^2}{5^2}}\) = \(\sqrt{\frac{9}{25}} = 3/5 = 0.6

The coordinates of the foci are F +(ae, 0) = (5(0.6), 0) = (+3, 0), and F'(-ae, 0) = (-3, 0).

Therefore, the coordinates of the foci of ellipse are (+3, 0), and (-3, 0).
Example 2: Find the foci of ellipse having the major axis of 12 units, minor axis as 8 units, and having the coordinate axes as the axis of the ellipse?

Solution:

The given values of the major axis and minor axes are 12 units and 8 units.

Hence we have 2a = 12, and 2b = 8, or a = 6, and b = 4 units respectively. Let us now calculate the eccentricity of the ellipse using the formula e =\(\sqrt {1 - \frac{b^2}{a^2}}\).

e =\(\sqrt {1 - \frac{4^2}{6^2}}\) =\(\sqrt {1 - \frac{4}{9}}\) = \(\sqrt {5}{9} \) = 0.74

The coordinates of the foci are F (+ae, 0) = (+6(0.714), 0) = (+4.2, 0), and F'(-ae, 0) = (-4.2, 0).

Therefore, the coordinates of the foci of the ellipse are (+4.2, 0), and (-4.2, 0).

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Practice Questions on Foci of Ellipse

Here are a few activities for you to practice. Select your answer and click the "Check Answer" button to see the result.

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FAQs on Foci of Ellipse

How Do You Find the Foci of Ellipse?

The foci of the ellipse can be found by knowing the value of the semi-major axis of the ellipse, and the value of eccentricity of the ellipse. For a standard equation of the ellipse x²/a² + y²/b² = 1, the semi-major axis length is 'a' units, and the value of eccentricity is e. Hence the coordinates of the two foci of the ellipse are F (+ae, o), and F' (-ae, 0).

What Is the Formula to Find Foci Of Ellipse?

The formula to find the foci of the ellipse can be understood from the equation of the ellipse. For an ellipse (x - h)²/a² + (y - k)²/b² = 1, the center of the ellipse is (h, k), and the coordinates of foci are F (+(h + a)e, k), and F'((h - a)e, k).

How Many Foci Does an Ellipse Have?

The ellipse has two foci located on the major axis of the ellipse.

Where Do We Use Foci Of Ellipse?

The foci of ellipse is used in finding the focal length, the equation of the focal chords, and the latus rectum. Further, the basic definition of the ellipse is based on the foci of the ellipse.

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