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
|The basic concepts of an ellipse involve:
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:
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?
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).
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.
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.
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
y = [−l / m]x + [n / m] is tangent to
n2 = m2b2 + l2a2