Foci of the ellipse

We are all aware of the rotation and revolution of the earth. We have learned that the earth can rotate in a circular or spinning movement around an imaginary line known as its axis. This movement is also exhibited by other planets and their moons. The term revolution is often used as another term for rotation and involves the movement of the moon around the earth and the earth around the sun. Now, this revolution of the earth around the sun occurs in a fixed path known as an elliptical orbit. This means that the path taken by the earth during its circular movement around the sun resembles the shape of a mathematical curve known as an ellipse. Thus, the motion of the earth around the sun is said to be elliptical.

Ellipse

In mathematical terms, an ellipse is defined as the locus of all set points in a plane fixed in such a way that the sum of the distances between the points is constant. The intersection of a cone along with a plane is given as the ellipse if there is no intersection with the cone’s base. The ellipse generally looks circular or oval. The fixed point where all the setpoints meet is known as the focus, denoted by S. The fixed line is known as the directrix, denoted by d and the constant ratio of the distances is known as the eccentricity, denoted by e. Based on the value of e, the conic shape can be determined.

• If the constant e = 1, this implies that the conic is a parabola
• If the constant e < 1, this implies that the conic is an ellipse
• If the constant e > 1, this implies that the conic is a hyperbola

Foci of an Ellipse

The ellipse generally consists of two fixed points known to be foci and their coordinate values can be given as  F(c, o) and F'(-c, 0). This implies that the distance between the foci is equal to 2c. Equation of the ellipse The general format for an elliptical equation represents an ellipse in the coordinate plane algebraically. This can be represented in two ways:

1. i) x²/a² + y²/b² = 1 where x-axis = transverse axis,  y-axis = conjugate axis
2. ii) x²/b² + y²/a² = 1 where y-axis = transverse axis,  x-axis = conjugate axis

Finding the Foci of the Ellipse

The foci of the ellipse play a major part in deriving the equation of the ellipse. They represent the two reference points that are equidistant from the origin and lie on the major axis of the ellipse. To calculate the foci of the ellipse, we need to know the values of the semi-major axis, semi-minor axis, and the eccentricity (e) of the ellipse. The  formula for eccentricity of the ellipse is given as e = √1−b²/a² Let us consider an example to determine the coordinates of the foci of the ellipse. Let the given equation be  x²/25 + y²/16 = 0. By comparing this equation with the general format, we get a =5 and b = 4. Let us start by calculating the value of eccentricity of the ellipse. We know that,  e = √1−b²/a² Therefore, e = √1−42/52  =  √1−9/25 = 3/5 = 0.6 So, the coordinates of the foci are given as: F +(ae, 0) = (5(0.6), 0) = (+3, 0) F'(-ae, 0) = (-5(0.6), 0) = (-3, 0) Hence, the coordinates of the foci of the ellipse are (+3, 0) and (-3, 0). Let us consider another example of finding various parts of the ellipse using respective formulae. Find out the following for the equation  x²/25 + y²/100 = 1. i) coordinates of the foci ii) coordinates of the vertices iii) the length of the major axis iv) the length of the minor axis v) the eccentricity and vi) the length of the latus rectum Solution The given equation x²/25 + y²/100 = 1 can be written as x²/52 + y²/102 = 1. So, b = 5, a = 10. ae = c = √100-25 = √75 = 5√3. Therefore,  the coordinates of the foci are (0,±5√3) The coordinates of the vertices are (0,±10) Length of major axis =2a=20 Length of minor axis =2b=10 e = c/a = 5 √3/10 = √3/2 Length of latus rectum = 2b²/a = 2*25/10 = 5

Conclusion

The ellipse is a closed circular-shaped structure present in a two-dimensional plane and covers a region in a 2D plane. This particular region covered or bounded by the ellipse is its area. The formula, different parts of the ellipse, and its fixed points called foci, along with examples, are elaborated in this article. These study material notes on foci can help us better understand this concept and be useful for students participating and preparing for various competitive exams.