pl help me to know the ellipse in detail 

for any conic section, the basic ingredients that needs to be known are:

1. fixed point S called the Focus

2. fixed line l called Directrix

3. moving point P.

 

The conic section differs based on the behaviour of the moving point in the plane with respect to S and l mentioned above.

Now to mention a 4 th ingredient called the "eccentricity", denoted as e , which is the constant ratio PS : PM where M is the foot of the perpendicular drawn from P to l.

 

In an Ellipse, this e is less than 1. An ellipse with centre C(0,0) having principal axes AA' as transverse (major) axis and BB' as conjugate (minor) axis, has two foci S(ae, 0) and S'(-ae,0),  vertices A (a, 0) and A' (-a, 0) and two directrices whose equations are x = -a/e and x = a/e. The line that passes through a focus perpendicular to AA' is called the Latus Rectum. The ellipse has two of these passing through S and S'.

 

A circle (x^2 + y^2 = r^2 : equation of circle having radius r) is a limiting case of an ellipse.

So, (x - ae)^2 + y^2 = e^2 ((a/e) - x)^2 which when simplified gives :

 

x^2/a^2  + y^2/(a^2(1 - e^2))  = 1.

 Now, if e <1 , we have (1 - e^2) > 0. So, substituting (a^2(1 - e^2)) as b^2 we have the standard form of equation of ellipse to be:

 

x^2/a^2  + y^2/b^2  = 1.

 

We have the parametric case as well which I shall explain if you need it.