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
Relationship between semi-major axis, semi-minor axis and the distance of the focus from the centre of the ellipse
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
$c^2= a^2 -b^2$
Eccentricity of ellipse
It is the ratio of the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse.
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
Standard equation of Ellipse with Foci on the x-axis and center as origin
$\frac {x^2}{a^2} +\frac {y^2}{b^2} =1$
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
Standard equation of Hyperbola when Foci lies on y-axis and center at origin
$\frac {x^2}{b^2} + \frac {y^2}{a^2} =1$
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
Important Observtion
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
Latus Rectum Of Ellipse
Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse
$l= \frac {2b^2}{a}=2a(1-e^2)$
Parametric Form of ellipse
$x = a cos \theta$ and $y = b sin \theta$ together represent the ellipse.
Position of a Point with Respect to a Ellipse
Let the Ellipse Equation $S ≡ \frac {x^2}{a^2} +\frac {y^2}{b^2} -1=0$
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 ($S_1 = 0$).
Equation of tangent of Ellipse
Equation of tangent at $P(x_1,y_1)$ in a Ellipse
$\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 of Ellipse
Normal to any point on the Ellipse is the line which passes through point and its perpendicular to the tangent at that point
Equation of Normal
$\frac {x -x_1}{x_1/a^2} =\frac {y -y_1}{y_1/b^2} $
Pair of Tangents of Ellipse
Two tangents, PQ and PR, can be drawn from a point $P(x_1,y_1)$ to the Ellipse $S ≡ \frac {x^2}{a^2} +\frac {y^2}{b^2} -1=0$
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.
Chord of Contact of Ellipse
The chord joining the two points of contact of tangents to a Ellipse drawn from any point P is called the chord of contact of P with respect to the given Ellipse.
Equation of Chord of Contact
$xx_1/a^2 +yy_1/b^2 =1$
