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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