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Ellipse Formula

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  • Last Updated : 01 Jun, 2022
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An ellipse is the locus of all points on a plane with constant distances from two fixed points in the plane. The fixed locations encircled by the curve are known as foci (singular focus). The constant ratio is the eccentricity of the ellipse and the fixed line is directrix. Eccentricity is an ellipse factor that shows elongation and is symbolized by the letter ‘e.’

Ellipse

 

The ellipse has an oval shape, and the area of an ellipse is specified by its major and minor axes. Ellipse area = πab, where a and b are the lengths of an ellipse’s semi-major and semi-minor axes. Ellipse is analogous to other parts of the conic section that are open and unbounded in shapes, such as parabola and hyperbola. In terms of locus, an ellipse is the set of all points on an XY plane whose distance from two fixed points (called foci) adds up to a constant value. The ellipse is a type of conic section formed when a plane cuts a cone at an angle with its base. A circle is formed when the plane intersects the cone parallel to its base.

What is Ellipse?

In geometry, an ellipse is a two-dimensional shape described along its axes. When a cone is intersected by a plane at an angle with respect to its base, an ellipse is created. There are two focal points. The total of the two distances to the focal point is always constant for all places along the curve. A circle is also an ellipse in which the foci are all at the same location, which is the circle’s center.

Properties of Ellipse:

  • Ellipse includes two focal points, also referred to as foci.
  • A fixed distance is referred to as a directrix.
  • The eccentricity of an ellipse ranges from 0 to 1. 0≤e<1
  • The whole sum of each distance from an ellipse’s locus to its two focal points is constant.
  • Ellipse has one major and one minor axis, as well as a center.

Components of an Ellipse:

  • Axis (Major and Minor) – Ellipses are distinguished by two axes running along the x and y axes:
    1. Major Axis: The major axis is the ellipse’s longest diameter, running through the center from one end to the other at the broadest part of the ellipse. 
    2. Minor Axis: The minor axis is the shortest diameter of an ellipse that crosses through the center at its narrowest point. Half of the major axis is the semi-major axis, and half of the minor axis is the semi-minor axis.
  • Eccentricity of the ellipse – The eccentricity of an ellipse is defined as the ratio of distances from the center of the ellipse to the semi-major axis of the ellipse.

e = c/a

where 

  • c is the focal length and 
  • a is the semi-major axis length.

Since c a, the eccentricity of an ellipse is always larger than 1.

Furthermore, c2 = a2 – b2.

As a result, eccentricity becomes:

e = √[(a2 – b2)/a2]

e = √[1-(b2/a2)]

Ellipse Formula 

Take a point P at one end of the major axis, as indicated. As a result, the total of the distances between point P and the foci is,

F1P + F2P = F1O + OP + F2P = c + a + (a–c) = 2a

Then, select a point Q on one end of the minor axis. The sum of the distances between Q and the foci is now,

F1Q + F2Q = √ (b2 + c2) + √ (b2 + c2) = 2√ (b2 + c2)

We already know that points P and Q are on the ellipse. As a result, by definition, we have

2√ (b2 + c2) = 2a

then √ (b2 + c2) = a

i.e. a2 = b2 + c2 or c2 = a2 – b2

The following is the equation for ellipse.

c2= a2 – b2

Standard equations for ellipse

The ellipse equation with its center at the origin and its major axis along the x-axis is:

x2/a2+y2/b2 = 1

where –a ≤ x ≤ a.

The ellipse equation with the center at the origin and the major axis along the y-axis is:

x2/b2+y2/a2 = 1

where –b  ≤ y ≤ b.

Sample Questions

Question 1: If the length of the semi-major axis is given as 10 cm and the semi-minor axis is 7 cm of an ellipse. Find its area.

Answer: 

Given, the length of the semi-major axis of an ellipse, a = 10 cm

Length of the semi-minor axis of an ellipse, b = 7 cm

We know the area of an ellipse using the formula;

Area = π x a x b

= π x 10 x 7

= 70 x π

Therefore Area = 219.91 cm2

Question 2: Define the major and minor axis of an ellipse?

Answer: 

 Ellipses are distinguished by two axes running along the x and y axes:

  1. Major Axis: The major axis is the ellipse’s longest diameter, running through the center from one end to the other at the broadest part of the ellipse. 
  2. Minor Axis: The minor axis is the shortest diameter of an ellipse that crosses through the center at its narrowest point. Half of the major axis is the semi-major axis, and half of the minor axis is the semi-minor axis.

Question 3: What are the equations for Ellipse?

Answer: 

The ellipse equation with its center at the origin and its major axis along the x-axis is:

x2/a2+y2/b2 = 1

where –a ≤ x ≤ a.

The ellipse equation with the center at the origin and the major axis along the y-axis is:

x2/b2+y2/a2 = 1

Question 4: Find the lengths for the major axis and minor axis of equation 7x2+3y2= 21

Answer: 

Given equation is  7x2+3y2= 21

dividing both sides by 21, we get

x2/3 + y2/7 = 1

we know ellipse standard equation 

   x2/b2+y2/a2 = 1 

as the foci lies on y-axis, for the above equation , the ellipse centered at origin and major axis on y-axis.

then ;

b2 = 3, which means b = 1.73

a2 = 7, which means a = 2.64

therefore

let the length of the major axis = 2a = 5.28

length of the minor axis = 2b =  3.46 

Question 5: What will be the area for an ellipse?

Answer:

The area of an ellipse is determined by the lengths of its minor and major axes.

Area of the ellipse = π.a.b

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