- An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is constant
- Foci: The two points are called the Foci
- Centre of Ellipse: The midpoint of the line segment joining the foci is called the centre of the ellipse
- Major axis: The line through the foci is called Major axis
- Minor axis: The line segment through the center and perpendicular to the Major axis
- Vertices: The end-point of Major axis is called the vertices

- We denote the length of the major axis by 2a, the length of the minor axis by 2b and the distance between the foci by 2c.
- Thus, the length of the semi major axis is a and semi-minor axis is b

For point P

Sum of the distance from two fixed points =2a

For point Q

Sum of the distance from two fixed points =$2 \sqrt {b^2 + c^2}$

Now it is should as per definition of ellipse

$2a=2 \sqrt {b^2 + c^2}$

Squaring both sides

$c^2= a^2 -b^2$

It is denoted by letter e

$e= \frac {c}{a}$

So c=ae

Therefore

$a^2e^2=a^2 -b^2$

or

$b^2=a^2(1-e^2)$

For an ellipse, 0

$\frac {x^2}{a^2} +\frac {y^2}{b^2} =1$

Here a> b

Foci can be found as (c,0) and (-c,0). It is on x-axis

$c^2 =a^2 -b^2$

Length of Transverse axis =2a

Vertices (a,0) and (-a,0)

Length of Conjugate axis =2b

$\frac {x^2}{b^2} + \frac {y^2}{a^2} =1$

a> b

Foci can be found as (0, c) and (0, -c). It is on y-axis

$c^2 =a^2 -b^2$

Length of Major axis =2a

Vertices (0, a) and (0, -a)

Length of Minor axis =2b

- Ellipse is symmetric with respect to both the coordinate axes since if (x, y) is a point on the ellipse, then (– x, y), (x, –y) and (– x, –y) are also points on the ellipse.
- The foci always lie on the major axis. The major axis can be determined by finding the intercepts on the axes of symmetry. That is, major axis is along the x-axis if the coefficient of $x_2$ has the larger denominator and it is along the y-axis if the coefficient of $y_2$ has the larger denominator

$l= \frac {2b^2}{a}=2a(1-e^2)$

A point $(x_1,y_1)$ lies outside the Ellipse if $S_1 ≡ \frac {x_1^2}{a^2} +\frac {y_1^2}{b^2} -1$ is positive ($S_1 > 0$).

A point $(x_1,y_1)$ lies inside the Ellipse if $S_1 ≡ \frac {x_1^2}{a^2} +\frac {y_1^2}{b^2} -1$ is negative ($S_1 < 0$).

A point $(x_1,y_1)$ lies one the Ellipse if $S_1 ≡ \frac {x_1^2}{a^2} +\frac {y_1^2}{b^2} -1$ is Zero ($S_1 = 0$).

$\frac {x^2}{a^2} +\frac {y^2}{b^2} =1 $

Equation of tangent

$xx_1/b^2 +yy_1/a^2 =1$

Another Form

The equation y = mx + c is a tangent to the ellipse if $c^2 = a^2m^2+ b^2$

Equation of Normal

$\frac {x -x_1}{x_1/a^2} =\frac {y -y_1}{y_1/b^2} $

The combined equation is

$(x^2/a^2 +y^2/b^2 -1)(x_1^2/a^2 +y_1^2/b^2 -1)= (xx_/a^2 +yy_1/b^2 -1 )^2$

This can be written like below to remember easily

$SS_1 = T^2$,

Where S = 0 is the equation of the circle, T = 0 is the equation of tangent at $(x_1, y_1)$, and S1 is obtained by replacing x by $x_1$ and y by $y_1$ in S.

Equation of Chord of Contact

$xx_1/a^2 +yy_1/b^2 =1$