Ellipse

What is an ellipse? The ellipse is one of the four traditional conic sections and is made by the slicing of a cone with a plane. The parabola, circle, and hyperbola are the other conic sections. The conic sections that planets and other celestial objects in the solar system trace out are also called ellipses, a fundamental subject in astronomy. In this article, we will be discussing ellipses in depth.

Definition of Ellipse

An ellipse is the set of all points on an XY-plane, whose distance from two fixed points equals a constant value. The two fixed points are known as 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 formed when a cone intersects a plane that does not intersect with the base of the cone. The focus is designated by S, the constant ratio ‘e’ is eccentricity, and the fixed-line is the directrix (d) of the ellipse.

Standard Equation of an Ellipse

When the ellipse’s centre is at (0,0) and the foci are on the x- and y-axes, the equation of an ellipse may be easily deduced. 

The equation of an ellipse is denoted by;

x2/a2 + y2/b2 and is equal to 1.

Terminologies Related to Ellipse

Ellipse diameter: Any straight line segment that passes through the centre of an ellipse and the line segment’s points lying on the ellipse is an ellipse’s diameter.

Eccentricity: (e<1). The distance of the focus from the ellipse’s centre is divided by the distance of one end of the ellipse from the ellipse’s centre. If the focus distance from the ellipse’s centre is ‘c,’ and the end distance from the centre is ‘a’, eccentricity e = c/a.

Major axis: The major axis is the ellipse’s longest diameter (typically represented as ‘a’), which runs through the centre from one end to the other.

Minor axis: The minor axis (denoted by ‘b’) is the ellipse’s shortest diameter, passing through the centre at its narrowest point.

Latus rectum: Latus rectum is a line drawn perpendicular to the ellipse’s transverse axis and passes through the ellipse’s foci. The length of the ellipse’s latus rectum is 2b2/a.

Transverse Axis: The transverse axis is the line that connects the two foci and the ellipse’s centre.

Area of an Ellipse

The radius of a circle can be used to compute its area, but the length of the minor and major axes determines the area of an ellipse.

We know that the area of a circle is equal to r2.

As a result, the ellipse’s area equals Major Axis x Minor Axis.

The variables a and b are the minor and major axis lengths, respectively.

Perimeter of an Ellipse

The perimeter of an ellipse is the whole length of its boundary and is measured in cm, m, ft, yd, and other units. The perimeter of an ellipse can be approximated using the following general formulas:

P ≈ π (a + b)

P ≈ π √[ 2 (a2 + b2) ]

P ≈ π [ (3/2)(a+b) – √(ab) ]

where,

a = length of semi-major axis

b = length of semi-minor axis

Properties of an Ellipse

• An ellipse is formed when a plane intersects a cone at its base angle.
• Every ellipse has two foci, a centre and a major and minor axis.
• The sum of the distances between any two foci on the ellipse produces a constant value.
• For all ellipses, the eccentricity value is less than one.

Real-life Application of Ellipse

• Ellipses can be used to represent planet orbits, satellite orbits, moon orbits, comet orbits, and the shapes of boat keels, rudders, and some aviation wings.
• A lithotripter generates sound waves to break up kidney stones using elliptical reflectors.

Conclusion

An ellipse is the set of all points on an XY-plane, whose distance from two fixed points equals a constant value. Some of the terminologies associated with ellipses are – ellipse diameter, eccentricity, major axis, minor axis, latus rectum, and transverse Axis. We have also discussed the properties of an ellipse and its real-life application. According to Kepler’s first law of planetary motion, each planet’s path is an ellipse with the Sun at one of its points. The ellipse’s reflection property is crucial in an elliptical pool because if the ball is hit through one focus, it will reflect off the ellipse and go into the hole at the other focus.