- A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point (not on the line) in the plane
- Directrix: The fixed line is called the directrix of the parabola
- Focus: the fixed-point F is called the focus
- Axis of Parabola: A line through the focus and perpendicular to the directrix is called the axis of the parabola.
- Vertex: The point of intersection of parabola with the axis is called the vertex.

Focus (a,0) a > 0 and directrix as x=-a

Let P (x,y) be any point on Parabola

Distance of P from Focus

$=\sqrt { (x-a)^2 + y^2}$

Distance of point P from Directrix

$=\sqrt {(x+a)^2}$

Since both of then them are equal

$\sqrt { (x-a)^2 + y^2}=\sqrt {(x+a)^2}$

$y^2=4ax$

Focus (-a,0) a> 0 and directrix as x=-a

Equation is

$y^2=-4ax$

Focus (0,a) a > 0 and directrix as y=-a

Equation is

$x^2=4ay$

Equation is

$x^2=-4ay$

(ii)When the axis of symmetry is along the x-axis the parabola opens to the

(a) right if the coefficient of x is positive,

(b) left if the coefficient of x is negative.

(iii)When the axis of symmetry is along the y-axis the parabola opens

(c) upwards if the coefficient of y is positive.

(d) downwards if the coefficient of y is negative

Latus Rectum of Parabola $y^2=4ax$ is 4a

A point $(x_1,y_1)$ lies outside the parabola if $S_1 ≡ y_1^2 - 4ax_1$ is positive ($S_1 > 0$).

A point $(x_1,y_1)$ lies inside the parabola if $S_1 ≡ y_1^2 - 4ax_1$ is negative ($S_1 < 0$).

A point $(x_1,y_1)$ lies one the parabola if $S_1 ≡ y_1^2 - 4ax_1$ is Zero ($S_1 = 0$).

(ii) $x=at^2$ and $y=2at$

$L=\frac {4}{m^2 } \sqrt {1+m^2} \sqrt {a(a-mc)}$

$y^2=4ax$

Equation of tangent

$yy_1=2a(x+x_1)$

Another line equation would be

$y=mx + \frac {a}{m}$

This is at the point ($am^2,-2am$)

Equation of Normal

$y-y_1 = \frac {- y_1}{a}(x-x_1)$

The combined equation is

$(y^2-4ax)(y_1^2-4ax_1)= (yy_1-2a(x+x_1))^2$

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

$yy_1-2a(x+x_1)=0$

The circle circumscribing the triangle formed by any three tangents to a parabola passes through the focus.

The orthocenter of any triangle formed by three tangents to a parabola $y^2 = 4ax$ lies on then directrix and has the coordinates $ –a, a(t_1 + t_2 + t_3 + t_1t_2t_3)$.

The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points.