ellipse
This line (joining the two foci) is called the major axis and a line drawn through the centre and perpendicular to the major axis is the minor axis. The endpoints are called vertices of the ellipse (Fig. 2).
ellipse
Also, we denote
The length of the major axis by ‘2a’
Length of the minor axis by ‘2b’
The distance between the foci by ‘2c’.
Browse more Topics under Conic Sections
Introduction to Conic Sections
Equation of Parabola
Equation of Hyperbola
Hence,
The length of the semi-major axis is ‘a’
Semi-minor axis is ‘b’ and
The distance of focus from the centre is ‘c’
ellipse
The Relationship Between ‘a’, ‘b’, and ‘c’
Take a look at the following diagram:
ellipse
As shown, take a point P at one end of the major axis. Hence, the sum of the distances between the point P and the foci is,
F1P + F2P = F1O + OP + F2P = c + a + (a – c) = 2a.
Next, take a point Q at one end of the minor axis. Now, the sum of the distances between the point Q and the foci is,
F1Q + F2Q = √ (b2 + c2) + √ (b2 + c2) = 2√ (b2 + c2)
We know that both points P and Q lie on the ellipse. Hence, by definition we have
2√ (b2 + c2) = 2a
Or, √ (b2 + c2) = a
i.e. a2 = b2 + c2 or c2 = a2 – b2
Special Cases
In the equation, c2 = a2 – b2, if we keep a fixed and vary the value of ‘c’ from ‘0-to-a’, then the resulting ellipses will vary in shape.
Case-I c = 0: When c = 0, both the foci merge together at the centre of the figure. Also, a2 becomes equal to b2, i.e. a = b. Hence, the ellipse becomes a circle.
Case-II c = a: When c = a, b = 0. Hence, the ellipse reduces to a line joining the two points F1 and F2.
Eccentricity: It is the ratio of the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse. It is denoted by ‘e’. Therefore, e = c/a.
Standard Equations of an Ellipse
When the centre of the ellipse is at the origin and the foci are on the x or y-axis, then the equation of the ellipse is the simplest. Here are two such possible orientations:
ellipse
Of these, let’s derive the equation for the ellipse shown in Fig.5 (a) with the foci on the x-axis. Let F1 and F2 be the foci and O be the mid-point of the line segment F1F2. Also, let O be the origin and the line from O through F2 be the positive x-axis and that through F1 as the negative x-axis.
Further, let the line drawn through O perpendicular to the x-axis be the y-axis. Let the coordinates of F1 be (– c, 0) and F2 be (c, 0) as shown in Fig.5 (a) above.
Derivation of the Equation
Now, we take a point P(x, y) on the ellipse such that, PF1 + PF2 = 2a
By the distance formula, we have,
√ {(x + c)2 + y2} + √ {(x – c)2 + y2} = 2a
Or, √ {(x + c)2 + y2} = 2a – √ {(x – c)2 + y2}
Further, let’s square both the sides. Hence, we have
(x + c)2 + y2 = 4a2 – 4a√ {(x – c)2 + y2} + (x – c)2 + y2
Simplifying the equation, we get √ {(x – c)2 + y2} = a – x(c/a)
We square both sides again and simplify it further to get,
x2/a2 + y2/(a2 – c2) = 1
We know that c2 = a2 – b2. Therefore, we have x2/a2 + y2/b2 = 1
Therefore, we can say that any point on the ellipse satisfies the equation:
x2/a2 + y2/b2 = 1 … (1)
Let’s look at the converse situation now. If P(x, y) satisfies equation (1) with 0 < c < a, then y2 = b2(1 – x2/a2)
Therefore, PF1 = √ {(x + c)2 + y2}
= √ {(x + c)2 + b2(1-x2/a2)}
Simplifying the equation and replacing b2 with a2 – c2, we get PF1 = a + x(c/a)
Using similar calculations for PF2, we get PF2 = a – x(c/a)
Hence, PF1 + PF2 = {a + x(c/a)} + {a – x(c/a)} = 2a.
Therefore, any point that satisfies equation (1), i.e. x2/a2 + y2/b2 = 1, lies on the ellipse. Also, the equation of an ellipse with centre of the origin and major axis along the x-axis is:
x2/a2 + y2/b2 = 1.
Note: Solving the equation (1), we get
x2/a2 = 1 – y2/b2 ≤ 1
Therefore, x2 ≤ a2. So, – a ≤ x ≤ a. Hence, we can say that the ellipse lies between the lines x = – a and x = a and touches these lines. Similarly, it can lie between the lines y = – b and y = b and touch those lines. Its equation {Fig. 5 (b)} is:
x2/b2 + y2/a2 = 1.
Hence the Standard Equations of Ellipses are:
x2/a2 + y2/b2 = 1.
x2/b2 + y2/a2 = 1.
Observations
An ellipse is symmetric with respect to both the coordinate axes. In simple words, if (m, n) is a point on the ellipse, then (- m, n), (m, – n) and (- m, – n) also fall on it.
The foci always lie on the major axis.
If the coefficient of x2 has a larger denominator, then the major axis is along the x-axis.
If the coefficient of y2 has a larger denominator, then the major axis is along the y-axis.
