Introduction to Ellipse:
An ellipse is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point in the same plane to its distance from a fixed straight line is always constant and is always less than unity.
The constant ratio is generally denoted by e and is known as the eccentricity of the ellipse.
If S is the focus, ZZ’ is the directrix and P is any point on the ellipse, then by definition SP/PM = e => SP = e. PM.
General Equation of an Ellipse: x^2/a^2 + y^2/b^2 = 1. This is an Ellipse Equation.
Let us understand how to find Equation of Ellipse:
To understand how to Find the Equation of an Ellipse We have, x^2/a^2 + y^2/b^2 = 1, where a > b. ……..(i)So, y = ± b/a sqrt(a^2 – x^2) ……..(ii) and x = ± a/b sqrt(b^2 – y^2) …….(iii).
We observe the following:
(a) Symmetry: For every value of x an equal and opposite values of y is found.
Similarly for every value of y an equal and opposite values of x is found as well. Thus the curve is symmetric about both the axes.
(b) Origin: The curve doesn’t pass from origin.
(c) Axes Intersection: The curve meets the x axis at y = 0. Putting y = 0 in (iii), we get x = ± a.
So the curve meets x-axis at A (a, 0) and A’ (-a, 0).
Putting x = 0 in (ii), we get y = ± b. So the curve meets y-axis at B(0, b) and B’ (0, -b). (d) Region: If x > a or x < -a, from (ii) we get imaginary values of y.
Therefore there is no part of the curve to the right of A or to the left of A’. If y > b or y < -b, from (iii) we get imaginary values of x.
Therefore there is no part of the curve above B(0, b) or below B’(0, -b). From (ii) we find that at x = 0, y = ± b and as x increases the value of y decreases and y = 0 at x = a.
Therefore the curve is a closed curve. With the help of the above facts and by joining some convenient points on the ellipse the general shape of the ellipse is x^2/a^2 + y^2/b^2 = 1.
Equations of ellipses in other forms: In the equation of the ellipse x^2/a^2 + y^2/b^2 = 1, if a > b or a^2 > b^2, then the major and minor axes lie along x-axis and y-axis respectively.
But if a < b or a^2 < b^2 (denominator of x^2 is less than that of y^2), then the major axis of the ellipse lies along the y-axis and is of length 2b and the minor axis along the x-axis and is of length 2a.
The coordinates of foci S and S’ are (0, be) and (0, -be) respectively. The equations of the directrices ZK and Z’K’ are y = ± b/e and eccentricity e is given by the formula a^2 = b^2 (1 – e^2) or e = sqrt(1 – a^2/b^2).
Ellipse Equation: If the centre of the ellipse is at the point (h, k) and the directions of the axes are parallel to the coordinate axes, then its equation is (x – h)^2/a^2 + (y – k)^2/b^2 = 1. If we shift the origin at (h, k) without rotating the coordinate axes, then x = X + h and y = Y + k. So, the equation of ellipses in reference to new origin, becomes X^2/a^2 + Y^2/b^2 = 1.
