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In mathematics, an ellipse is a set of points in a plane whose distance from a fixed point has a constant ratio of ‘e’ to the distance from a fixed line (less than 1). The ellipse is a part of the conic section, which is formed when a cone intersects a plane that does not intersect the base of the cone. The focus is denoted by S, the constant ratio ‘e’ is known as the eccentricity, and the fixed line is known as the directrix (d) of the ellipse.
Definition of Ellipse
An ellipse is a collection of points in a plane whose distances from two fixed points add up to a fixed value. The ellipse’s 2 fixed points are known as the foci.
Parts of an Ellipse
Focus: There are two foci. The foci (plural: focus) are always located along the major axis.
Center: The centre is the point where the two foci meet. At this point, the major and minor axes meet at a 90° angle.
Major axis: The distance between the end vertices is known as the major axis. The major axis is divided into two equal halves by the centre. Each half is referred to as a semi-major axis or major radius, and is denoted by the letter ‘a.’
Minor axis: The distance between the end co-vertices is known as the minor axis. The minor axis is divided into two equal halves by the centre. Each half is referred to as a semi-minor axis or minor radius, and is denoted by the letter ‘b.’
Vertex: The point where the ellipse intersects the major axis is called the vertex. In other words, the vertices are the two extreme points that make up the major axis.
Co-vertex: The point where the ellipse intersects the minor axis is known as the co-vertex. In other words, the co-vertices are the two extreme points that make up the minor axis.
Equation of Ellipse
An ellipse in the coordinate plane is represented algebraically using the general equation of an ellipse. An ellipse’s equation can be written as,
Area of Ellipse
The area of an ellipse in two dimensions is the area or region covered by the ellipse. An ellipse’s area is measured in square units such as in2, cm2, m2, yd2, ft2, and so on. Ellipse is a two-dimensional shape formed by connecting all points on the plane that are at a constant distance from two fixed points. The fixed points are known as ellipse foci.
The 2 foci are F1 and F2. Because an ellipse is not a perfect circle, the distance between its centre and the circumference’s points is not constant. An ellipse, then, has two radii. The major axis of the ellipse is the longest chord in the ellipse. The chord that is perpendicular to the major axis is known as the minor axis.
Ellipse is the locus of all points on a plane whose sum of distances between two fixed points is constant. For example, the points P1 and P2 are positioned so that the sum of point P1’s distances from the fixed points F1 and F2 equals the sum of point P2’s distances from the fixed points F1 and F2. That is, if we join all of the points P1, P2, P3, and so on, we will get an ellipse.
Given the lengths of the major and minor axes, the area of an ellipse can be calculated using a general formula. The formula for calculating the area of an ellipse is,
Area of ellipse = π a b
Where,
a = semi-major axis length
b = semi-minor axis length
Conclusion
The set of all points (x,y) in a plane where the sum of their distances from two fixed points is a constant is called an ellipse. Each fixed point is referred to as a focus (plural: foci). With the foci and vertices of an ellipse’s coordinates. One of the four classic conic sections created by slicing a cone with a plane is the ellipse. The parabola, circle, and hyperbola are the others. Celestial objects in periodic orbits around other celestial objects all trace out ellipses, which is why the ellipse is so important in astronomy.
