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:
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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?
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
Solution:
y = [−l / m]x + [n / m] is tangent to
Or
n2 = m2b2 + l2a2