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Table of Content

Ellipses Equations


Ellipses are the set of points where the sum of the distances from two foci to the given point is constant. Ellipse is the locus of a point which moves such as the ratio of its distance from focus to its distance from the directrix is less than 1.

Equation of the ellipse is in the form x^2/a^2 + y^2/b^2 = 1. S is the focus and l is the directrix, draw SZ perpendicular to l. on SZ mark the points A and A' such that SA/AZ=SA'/A'Z= e/1, therefore SA=eAz, let it be example 1. And SA'=eA'Z , let it be the example 2. Now bisect AA' at C. take C as the origin, CS produced as X-axis and CY perpendicular to CS as Y- axis. Let P(x,y) be any point on the ellipse.

Join mathematicsPS, draw Pm perpendicular to the directrix, PN perpendicular to X- axis. Take CA= CA' = a. Adding the example 1 and 2, we get SA + SA' = e(AZ + A'Z) that is AA'=e[(CZ-CA)+(CA'+CZ)],that is 2a =e(2CZ)(because CA =CA'),that is a = e times CZ therefore CZ = a/e. This means that the coordinates of z =(a/e,0).Now subtracting equation 1 from 2 example. That is (2)-(1),SA'-SA=e(A'Z-AZ),that is (CS+CA')- (CA-CS)= e(AA').that is ,2CS =e(2a) that is CS=Ae and this means that the coordinates of S =(ae,0),and by using the distance formula, PS= the whole squared root of (x-ae)^2 + (y-0)^2.looking into figure, PM= Nz = CZ-CN =a/e-x ,here because CN=x),thus PM= a/e-x. Since P is the point on the ellipse, PS/PM =e. Therefore PS=e times PM. That is under the whole root squared (x-ae)^2+(y-0)^2 =e(a/e-x) ,then Squaring both the sides and simplifying, that is x^2+A^2 times e^2-2xae+y^2=a^2+e^2 times x^2-2aex.thus after simplifying, we get x^2/a^2 + Y^2/a^2(1-e^2) =1 because e<1,a^2(1-e^2) is positive. Thus, take a^2(1-e^2)=b^2.thus our given formula is proved.

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