Ellipse

This shape belongs to the conic section and has traits with circles in terms of size, shape, and other qualities. Unlike a circle, which is round in form, an ellipse has an oval shape. The shape of an egg in two dimensions, as well as the running track in a sports stadium, are two fundamental examples of the ellipse that we encounter in our daily lives. 

Ellipse meaning

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 two fixed points are known as the foci.

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 portion 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 designated by S, the constant ratio ‘e’ is known as the eccentricity, and the fixed line is known as the directrix (d) of the ellipse.

Ellipse examples

An ellipse in the coordinate plane is represented algebraically using the generic equation of an ellipse. An ellipse’s equation can be written as,

x²/a²+y²/b²=1

Parts of Ellipse

• Focus:- The coordinates of the two foci on the ellipse are denoted by the letters F(c, o) and F’ (-c, 0). A consequence of this is that the distance between the foci is the same as 2c. 
• Minor Axis:- The minor axis of the ellipse is 2b units in length, and the end vertices of the minor axis are (0, b) and (0, -b), respectively, at the origin (0, b). 
• Major Axis:- A unit is equal to 2a units in length, and the main axis’ end vertices are (a, 0), (-a, 0), respectively, at the beginning and end of the ellipse. 
• Centre:- The centre of the ellipse is defined as the midpoint of the line connecting the two points of interest. 
• Transverse Axis:- The transverse axis is the line that runs through the ellipse’s centre and links the two foci. 
• Conjugate Axis:- It is a line that runs through the centre of the ellipse and is perpendicular to the transverse axis that is known as the conjugate axis. 
• Latus Rectum:- It is a line that is drawn perpendicular to the ellipse’s transverse axis and goes through the ellipse’s foci that is known as the latus rectum. The length of the ellipse’s latus rectum is equal to 2b2/a. 
• Eccentricity:- The distance between the focus and the ellipse’s centre divided by the distance between one end of the ellipse and the ellipse’s centre yields the radius of the ellipse. If the focus distance from the ellipse’s centre is ‘c,’ and the end distance from the centre is ‘a,’ the eccentricity of the ellipse is equal to the product of the focus distance and the end distance from the centre. 

Standard Equation of Ellipse

There are two standard ellipse equations. The transverse and conjugate axes of each ellipse are used to derive these equations. The ellipse’s standard equation: x²/a²+y²/b²=1

The x-axis is the transverse axis, while the y-axis is the conjugate axis. In addition, another classic ellipse equation is: x²/b²+y²/a²=1.

Eccentricity of an Ellipse

The eccentricity of an ellipse is defined as the relationship between the distance between the focus and the centre of the ellipse and the distance between one end of the ellipse and the centre of the ellipse. 

Eccentricity of an ellipse:- e=c/a= √1-b

Latus Rectum 

Latus rectum of an ellipse is defined as the line drawn perpendicular to the transverse axis of the ellipse and passing through the foci of the ellipse, as shown in the figure below. The following is the formula for determining the length of the latus rectum of an ellipse: 

L = 2b2/a

Conclusion

Ellipse curves are very essential in the mathematical sciences because they are so flexible. In the case of planets, for example, their orbits around the sun are elliptical. The use of ellipses is vital in engineering, architectural, and machine design drawings for two reasons. Beginning with the fact that every circle seen at an angle seems to be an ellipse, As a result of reading this article, we learned about many new concepts that are linked to elliptical equations, such as focus, eccentricity, the transverse axis, and others.