Derivation of Ellipse Equation
The 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 centre.
The image given below shows the foci F1 and F2 of the ellipse and points P1, P2, and P3 are points on the circumference of the ellipse.
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 centre of the ellipse is the minor axis. The endpoints P and Q as shown are known as the vertices which represent the intersection of major axes 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.
Proof:
Let’s take P and Q as the endpoint of the major axis and points R and S at the end of the minor axis and O as the centre of the ellipse.
Distances of Q from F1 is F1Q and Q to F2 is F2Q and their sum is F1Q + F2Q and F1Q + F2Q = F1O + OQ + F2Q
c + a + a – c = 2a
Sum of distances from point R to F1 is F1R + F2R
F1R + F2R = √(b2 + c2) + √(b2 + c2)
= 2√(b2 + c2)
By definition of the Ellipse,
2√(b2 + c2) = 2a
a = √(b2 + c2)
a2 = b2 + c2
c2 = a2 – b2
Ellipse
An ellipse is the locus of all points on a plane with constant distances from two fixed points in the plane. The fixed points encircled by the curve are known as foci (singular focus). The constant ratio is the eccentricity of the ellipse and the fixed line is the directrix. In this article, we will learn about the ellipse in detail.
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