An ellipse is the locus of all those points in a plane such that the sum of their distances from two fixed points in the plane is constant. The fixed points are known as the foci (singular focus) of the ellipse.
The above figure represents an ellipse such that P1F1 + P1F2 = P2F1 + P2F2 = P3F1 + P3F2 is a constant. This constant is always greater than the distance between the two foci. When both the foci are joined with the help of a line segment then the mid-point of this line segment joining the foci is known as the center, O represents the center of the ellipse in the figure given below:
The line segment passing through the foci of the ellipse is the major axis and the line segment perpendicular to the major axis and passing through the center of the ellipse is the minor axis. The end points A and B as shown are known as the vertices which represent the intersection of major axis with the ellipse. ‘2a’ denotes the length of the major axis and ‘a’ is the length of the semi-major axis. ‘2b’ is the length of the minor axis and ‘b’ is the length of the semi-minor axis. ‘2c’ represents the distance between two foci.
Let us consider the end points A and B on the major axis and points C and D at the end of the minor axis.
The sum of distances of B from F1 is F1B + F2B = F1O + OB + F2B (From the above figure)
⇒ c + a + a – c = 2a
The sum of distances from point C to F1 is F1C + F2C
⇒ F1C + F2C = √(b2 + c2) + √(b2 + c2) = 2√(b2 + c2)
By definition of ellipse
2√(b2 + c2) = 2a
a = √(b2 + c2)
⇒ a2 = b2 + c2
c2 = a2 – b2
- If c = 0 then F1 and F2e. both foci merge together with center of ellipse. Also a2 becomes equal to b2, i.e. a = b so now we get a circle in this case.
- If c = a then b becomes 0 and we get a line segment F1F2.
The eccentricity of the ellipse:
The ratio of distances from the center of the ellipse from either focus to either of the vertices of the ellipse is defined as the eccentricity of the ellipse.
Eccentricity of ellipse, e = c/a
Since c ≤ a the eccentricity is always greater than 1 in the case of an ellipse.
The simplest method to determine the equation of an ellipse is to assume that center of the ellipse is at the origin (0, 0) and the foci lie either on x- axis or y- axis of the Cartesian plane as shown below:
Both the foci lie on x- axis and center O lies at the origin.
Let us consider the figure (a) to derive the equation of an ellipse. Let the coordinates of F1 and F2 be (-c, 0) and (c, 0) respectively as shown. Let us consider a point P(x, y) lying on the ellipse such that P satisfies the definition i.e. the sum of distances of P from F1 and F2 in the plane is a constant 2a.
⇒ PF1 + PF2 = 2a – – – (1)
Using distance formula the distance can be written as:
Squaring and simplifying both sides we get;
Now since P lies on the ellipse it should satisfy equation 2 such that 0 < c < a.
Therefore the equation of the ellipse with center at origin and major axis along the x-axis is:
where –a ≤ x ≤ a.
Similarly, the equation of the ellipse with center at origin and major axis along the y-axis is:
where –b ≤ y ≤ b.
Latus Rectum: The line segments perpendicular to the major axis through any of the foci such that their end points lie on the ellipse are defined as the latus rectum.
The length of latus rectum is 2b2/a.
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