Ellipse

What is an Ellipse?

• An ellipse can be defined as the locus of all those points in a plane such that the sum of their distances from any two given fixed points in the plane is constant.

•  The foci (singular focus) are the fixed points, which are surrounded by the curve.

• The shape of the ellipse is in an oval shape and the major axis and minor axis define its area.

• A factor of the ellipse is known as the eccentricity that demonstrates the elongation of it and is denoted by the variable ‘e’.

Ellipse is similar to other parts of the conic section, for example the parabola and hyperbola, which are open in the shape and unbounded. The ellipse is generally defined by its equation, which we are going to learn in this article, along with the formula of area of the ellipse.

Definition of Ellipse

If we speak in terms of locus an ellipse can be defined as, it is the set of all points on the XY-plane, whose distance from any two fixed points (that is known as foci)and basically adds up to a constant value.

A circle can also be an ellipse, where the foci are at the same point and that is the center of the circle.

An ellipse is defined by its two-axis along x and y-axis: Minor Axis and the Major axis.

What is the Major and Minor Axis?

The major axis is known to be the longest diameter of the ellipse, going through the center from one end to the other end, at the broad part of the ellipse. The minor axis is known as the shortest diameter of ellipse, which crosses through the center at the narrowest part.

What is the Eccentricity of the Ellipse?

The ratio of distances from the center of the ellipse from either the focus to either of the vertices of the ellipse is known as the eccentricity of the ellipse.

The eccentricity of ellipse is equal to, e = c/a

Since the value of c ≤ a the eccentricity (e) is always greater than the value of 1 in the case of an ellipse.

Equation of an Ellipse

When the centre of the ellipse is at the origin that is equal to (0,0) and the foci are on the x-axis and  the y-axis, then the equation of an ellipse  can be easily derived -

The equation of the ellipse is denoted by;

x2/a2 + y2/b2 is equal to 1.

Things You Need to Know About an Ellipse

Diameter of Ellipse – Diameter of an ellipse can be defined as any straight line segment that passes through the center of an ellipse and the line segment’s points lie on the ellipse.

Linear Eccentricity (c) – Linear eccentricity can be defined as the distance from the focal point to the center of the ellipse.

Eccentricity of Ellipse (e) – Eccentricity of an ellipse can be defined as the ratio of the linear eccentricity c to the length of the semi-major axis that is denoted by a. The eccentricity of an ellipse can always be denoted as 0 < e < 1, the eccentricity of the circle is e = 0, the eccentricity of the parabola is e is equal to 1 and the eccentricity of the hyperbola is e > 1.

E = c/a

Radius of an ellipse R – The radius of an ellipse R can be defined as the distance from ellipse the center to point denoted by Mn at ellipse.

 Radius (R)  = ab which is further equal to b √a2sin2φ + b2cos2φ √1 - e2cos2φ

e = eccentricity and φ – is the angles within the radius (R) and major axis A1A2.

Focal parameter of ellipse (p) - Focal parameter of ellipse is the focal radius that perpendicular to major axis:

 Focal Parameter(p) = b2 a

Directrix of ellipse (1 - k ) can be defined as a line parallel to the minor axis and no touch to the ellipse. The distance from any point M on the ellipse to the given focus F is a constant fraction of that points perpendicular distance to the directrix, that results in the equality p/e.

Why is an Ellipse Important?

The ellipse is one of the four classic conic sections that is generally created by slicing a cone with a plane. The other conic sections are the parabola, the circle, and the hyperbola. The ellipse is a very important concept in astronomy as celestial objects in periodic orbits around other celestial objects all trace out the conic section known as ellipses.

Area of Ellipse

Area of the circle can be calculated based on its radius, but the area of an ellipse depends on the length of the minor axis and major axis.

We know that the area of the circle = πr2

Therefore, area of the ellipse = π x Major Axis x Minor Axis

Where the variables a and b are the length of the minor axis and major axis

Questions to Be Solved

Question 1) If the length of the major axis is equal to 7cm and the minor axis is equal to 5cm of an ellipse. Find the area of the ellipse.

Solution) Given, the length of the major axis of an ellipse is equal to 7cm

Length of the minor axis of an ellipse is equal to 5cm

By the formula of area of an ellipse, we know that;

Area of the ellipse = π x major axis x minor axis

Area of the ellipse = π x 7 x 5

Area of the ellipse = 35 π

We know that π = 22/7

Area = 35 x 22/7

Therefore Area of the ellipse = 110 cm2