Let `S` be the focus & `ZM` is the directrix of an ellipse. Draw perpendicular from `S` to the directrix which meet it at `Z`. A moving point is on the ellipse such that

`PS = e PM`

then there is point lies on the line `SZ` and which divide `SZ` internally at `A` and externally at `A'` in the ratio of `e: 1`.

therefore `SA=eAZ`..........................(i)

`SA'=eA'Z`..................................(ii)

Let `A A' = 2a` & take `C` as mid point of `A A'`

`:. CA = CA' = a`

Add `(i)` & `(ii)`

`SA + SA' = e (AZ+A'Z)`

`=> A A' = e[CZ - CA + CA' + CZ]`

`2a = 2eCZ`

`=> CZ=a/e`.....................(iii)

Subtract `(ii)` & `(i)`, we get

`SA' - SA = e (A'Z - AZ)`

`=> (CA' + CS) - (CA - CS)`

`= e [(CA' + CZ) - (CZ - CA)]`

`2CS = 2e CA`

`:. CS = a e` .........................(iv)

Result `(iii)` & `(iv)` are independent of axis.

Consider `CZ` line as `x`-axis, `C` as origin & perpendicular to this line & passes through `C` is considered as `y`-axis. Let `P(x, y)` is a moving point, then By definition of ellipse.

`PS = ePM (PS)^2 = e^2 (PM)^2`

`=> (x-ae)^2+(y-0)^2=e^2(a/e -x)^2 => (x-a e)^2+y^2=(a-ex)^2`

`=> x^2+a^2e^2-2xae+y^2=a^2+e^2x^2-2xae => x^2(1-e^2)+y^2=a^2(1-e^2)`

`=> x^2/a^2+y^2/(a^2(1-e^2)=1 ` or ` x^2/a^2+y^2/b^2=1`

where `b^2=a^2(1-e^2)`

(1) By the symmetry of equation of ellipse, if we take second focus `S'( - ae, 0)` & second directrix `x =- a/e` & perform same calculation, we get same equation of ellipse, therefore there are two focii & two directrix of an ellipse. The two focii of ellipse are `(ae, 0)` and (- ae, 0) and the two corresponding directrices are lines `x = -a/e` and `x =- a/e` . If focus of the ellipse is taken as `(ae, 0)`, then corresponding directrix is `x=-a/e` and if focus is `(- ae, 0)`, then corresponding directrix is `x = -a/e`

(2) If equation of directrix is `px + qy+ r= 0` & focus is `(h, k)` then its equation will be

`PS^2 = e^2 PM^2`

`(x-h)^2+(y-k)^2=e^2.((px+qy+r)/(sqrt(p^2+q^2)))^2`

(3) Distance between foci `SS' = 2ae` & distance between directrix `Z Z' = 2 a/e`

(4) Degree of flatness of an ellipse is also called on eccentricity & written as

`e=(CS)/(CA)=(text(Distance from centre to focus))/(text(Distance from centre to vertex))`

If `e -> 0 => b -> a =>` foci becomes closer & move towards centre and ellipse becomes circle.

If `e-> 1 =>b-> 0 =>` ellipse get thinner & thinner

(5) Two ellipse are said to be similar if they have same eccentricity.

(6) Distance of focus from the extremity of minor axis is equal to `'a'` because `a^2e^2 + b^2 = a^2`

(7) Let `P(x, y)` be any point on the ellipse .

`x^2/a^2+y^2/b^2=1`

`:. y^2/b^2=1-x^2/a^2`

`=> y^2/b^2=(((a-x)(a+x))/a^2) => (PN^2)/b^2= (AN . A' N)/a^2`

`=> (PN^2)/(AN .A'N) =b^2/a^2`

A circle described on major axis as diameter is called the auxiliary circle of the given ellipse & its equation is

`x^2+y^2=a^2` ..........................(1)

and given ellipse is `x^2/a^2+y^2/b^2=1` ....................(2)

Let `Q` be a point on the auxiliary circle `x^2 + y^2 = a^2` then line through `Q` and perpendicular to `x`-axis meet the ellipse
at `P` then `P` and `Q` are called the CORRESPONDING POINTS on the ellipse & the auxiliary circle respectively. Here `angle QOA = theta ` is called the ECCENTRIC ANGLE of the point `P` on the ellipse `(0 le theta < 2 pi)`.

Since ` Q` lie on the circle therefore `Q(a cos theta, a sin theta)`

So coordinate of `P(a cos theta, y)`, which satisfy the equation of ellipse.

`:. (a^2 cos ^2 theta)/a^2+y^2/b^2=1 =>y= b sin theta`

`:.` coordinate of `P` will be `(a cos theta, b sin theta)` and this is called parametric equation of ellipse.

The equations `x = a cos theta` & `y = bsin theta` together represent the ellipse `x^2/a^2+y^2/b^2=1`.

Where `theta` is an eccentric angle of point `P`

We observe that `(l(PN))/(l(QN))=b/a= (text(Semi minor axis))/(text(Semi major axis))`

Hence " If from each point of a circle perpendiculars are drawn upon a fixed diameter then the locus of the points dividing these perpendiculars in a given ratio is an ellipse of which the given circle is the auxiliary circle" . This another defintion of ellipse.

There are two standard forms of ellipse with centre at the origin and axes along coordinate axes. The foci of the ellipse are either on the `x`-axis or on they-axis.

1. Major Axis Along `x`-axis :
The equation of this type of ellipse is of the form `x^2/a^2+y^2/b^2=1` , where `a>b>0` and `b =a sqrt(1-e^2)` .

For this ellipse :

(i) Major axis is `2a`

(ii) Minor axis is `2b`.

(iii) Centre is `(0, 0)`

(iv) Vertices are `(pm a, 0)`

(v) Foci are `(pm a e, 0)`

(vi) Equation of directrices are `x= pm a/e`

(vii) Equation of major axis is `y =0`

(viii) Equation of minor axis is `x = 0`

(ix) Length of latus rectum `= (2b^2)/a`

(x) Extremity of latus rectum is `(+-a e,+-b^2/a)`

2. Major Axis Along `y`-axis :
The equation of this type of ellipse is of the form `x^2/a^2+y^2/b^2=1`. where `0 < a < b` and `a=bsqrt(1-e^2)`

For this ellipse:

(i) Major axis is `2b`

(ii) Minor axis is `2a`.

(iii) Centre is `(0, 0)`

(iv) Vertices are `(0, pm b)`

(v) Foci are `(0, pm be)`

(vi) Equation of directrices are `y = pm b/e`

(vii) Equation of major axis is `x = 0`

(viii) Equarion of minor axis is `y =0`

(ix) Length of latus rectum `=(2a^2)/b`

(x) Extremity of latus rectum is `( pm a^2/b, pm be)`

If the equation of the ellipse is given as `x^2/a^2+y^2/b^2= 1` & nothing is mentioned then the rule is to assume that `a> b`.

`x^2/a^2+y^2/b^2=1` is given ellipse

Let `P(x, y)` be any point on the ellipse, then `PM = y` & `PN = x`

`:.` above equation can be written as `(PN)^2/a^2 +(PM)^2/b^2=1`

From above we conclude that if perpendicular distances `p_1` & `p_2` of a moving point `P(x, y)` from two mutually perpendicular straight lines `L_1 = lx + my+n_1 = a` & `L_2 = mx - ly+n_2 = 0` respectively then equation of ellipse in the plane of line will be

`(p_2^2)/a^2+(p_1^2)/b^2=1`

`=> ((mx-ly+n_2)/(sqrt(l^2+m^2)))^2/a^2+((lx-my+n_1)/(sqrt(l^2+m^2)))^2/b^2=1`