Basic Concepts of Ellipse

Ellipse is one of the conic sections which is obtained by the intersection of a right circular cone. An ellipse is the locus of a point traversing in a plane, such that the ratio of its distance from the fixed point and the line is a constant. It is always less than one. The eccentricity of an ellipse is e < 1. The general equation of a conic is ax² + 2hxy + by² + 2gx + 2fy + c = 0. 

For example, If an egg is sliced in an oblique way, a curve can be seen by its edge.

The earth’s movement around the sun traces a similar but a bigger curve. 

These curves can be termed as an ellipse.

General Definition of Ellipse

An ellipse is the locus of a point traversing in a plane such that the ratio of its distance from the fixed point and the line is a constant. It is always less than one. The eccentricity of an ellipse is e < 1. Read More

Notations and Standard Equation Of Ellipse

The representation of the different parts of the ellipse are as follows:

• The length of the major axis by ‘2a’.
• The length of the minor axis by ‘2b’.
• The distance between the foci by ‘2c’.
• The distance of focus from the centre is ‘c’.
• The relation between ‘a’, ‘b’, and ‘c’ is a² = b² + c² or c² = a² – b².

The standard equations of an ellipse is given by:

• $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
• $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$

Solved Ellipse Problems

Example 1: If the latus rectum of an ellipse is equal to half of its minor axis, then what is its eccentricity?

Solution:

2b² / a = b

b / a = 1 / 2

b² / a² = 1 / 4

Hence e = √[1−b² / a²] = √3 / 2

Example 2: Find the equation of the ellipse whose centre is at origin and which passes through the points (3, 1) and (2, 2).

Solution:

Since it passes through (3, 1) and (2, 2), so

Hence required equation of ellipse is 3x² + 5y² = 32

Example 3: An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, then find the necessary length of the string and the distance between the pins in cm.

Solution:

Given 2a = 6, 2b = 4

i.e. , a = 3, b = 2

e² = 1 − b² / a² = 5 / 9

e = √5 / 3

Distance between the pins = 2*a*e = 2√5 cm

Length of string = 2*a + 2*a*e = 6 + 2 √5 cm

Example 4: The locus of a variable point whose distance from (2, 0) is 2 / 3 times its distance from the line x = −[9 / 2] is an ellipse or a parabola. Check your answer.

Solution:

Let point P be (x₁ , y₁)

Consider √[(x₁+ 2)²+ y₁²] = [2 / 3] (x₁ + 9 / 2)

(x₁ + 2)² + y₁² = [4 / 9] [(x₁ + 9 / 2)²]

9 [x₁² + y₁² + 4x₁ + 4] = 4 (x₁² + 81 / 4 + 9x₁ )

5x₁² + 9y₁² = 45

x₁² / 9 + y₁² / 5 = 1

Locus of (x₁ ,y₁) is x² / 9 + y² / 5 = 1, which is equation of an ellipse.

Example 5: What is the condition for the line lx + my− n=0 to be a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$?

Solution:

y = [−l / m]x + [n / m] is tangent to $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, if

Or

n² = m²b² + l²a²