(1) Axis : A straight line passes through the focus and perpendicular to the directrix is called the axis of parabola. For the parabola `y^2=4ax`, `x`-axis is the axis.
Since equation has even power of `y` therefore the parabola is symmetric about `x`-axis i.e. about its axis.

(2) Vertex: The point of intersection of a parabola and its axis is called the vertex of the Parabola. For the parabola `y^2 = 4ax , 0(0, 0)` is the vertex.
The vertex is the middle point of the focus and the point of intersection of axis and directrix.

(3) Focal Distance : The distance of any point `P (x, y)` on the parabola from the focus is called the focal length (distance) of point `P`.
The focal distance of `P =` the perpendicular distance of the point `P` from the directrix.

(4) Double Ordinate: The chord which is perpendicular to the axis of Parabola or parallel to Directrix is called double ordinate of the Parabola.

(5) Focal Chord : Any chord of the parabola passing through the focus is called Focal chord.

(6) Latus Rectum: If a double ordinate passes through the focus of parabola then it is called as latus rectum. The extremities of the latus rectum are `L (a, 2a)` and `L'(a, - 2a)`. Since `LS = L'S = 2a`, therefore length of the latus rectum `LL' = 4a`.

(7) Parametric Equation of Parabola: The parametric equation of Parabola `y^2 = 4ax` are `x = at^2, y= 2at.`
Hence any point on this parabola is `(at^2, 2at)` which is also called as `'t'` point.

lf the equation of a parabola is either in the form `x =l y^2 + my + n` or `y = l x^2+ mx + n` then it can be reduced into generalised form. For this we change the given equation into the following forms-

`(y - k)^2 = 4a (x - h)` or `(x - h)^2 = 4a (y - k)`

And then we compare from the standard equation of parabola to find all its parameters.

(A) When the equation of parabola is : `(y -k)^2 = 4a(x - h)`

`(y -k)^2 = 4a(x - h)`................(i)

Equation (i) is of the form `Y^2 = 4aX`

where `Y = y - k` and `X = x - h`

(1) Axis of parabola is `Y = 0`, i.e., ` y - k = 0 => y = k`

(2) Coordinates of vertex of parabola are given by

`X=0` and `Y=0`

i.e, `x-h=0` and `y-k=0`

`:.` Vertex is `(h,k)`

(3) Tangent at the vertex to parabola (i) is given by

`X=0`, i. e. , `x - h = 0`

Therefore, tangent at the vertex is `x = h` .

(4) Coordinates of focus of parabola are given by

`X = a` and `Y = 0`

i.e. by `x - h = a` and `y - k = 0`

`:.` Focus of parabola is `(a + h, k)`.
(5) Equation of directrix of parabola is

`X =-a`

i.e., `x - h = -a`

Therefore, directrix of parabola is `x = h - a`

(6) Length of latus rectum of parabola is `|4a |`

(7) Coordinates of ends of latus rectum of parabola are given by

`X=a` & `Y=+-2a`

i.e., by `x-h=a` , `y-k=+-2a`

i.e., coordinate of latus rectum is `(a+h,k+-2a)`.

(8) Parametric equation is `x = h + at^2` and `y = k + 2at`.

(B) When the equation of parabola is :

`(x - h)^2 = 4a (y - k)` ........................(i)

Equation (i) is of the form `X^2 = 4aY`

where `X = x - h` and `Y = y - k`

(1) Axis of parabola is `X = 0`, i.e. , `x - h = 0`

(2) Coordinates of vertex of parabola is given by

`X = 0` and `Y = 0`

i.e. , by `x - h = 0` and `y - k = 0`

`:. x=h` and `y=k`

Hence vertex of parabola is `(h , k)`

(3) Equation of tangent at the vertex to parabola is

`Y = 0` i.e. , `y - k = 0`

or `y = k`

(4) Coordinates of focus of parabola are given by

`X=0` and `Y=a`

i.e., by `x-h=0` and `y-k=a`

`:.` Focus of Parabola is `(h,k+a)`.

(5) Equation of directrix of parabola (i) is given by

`Y =-a` or `y - k = - a` or `y = k - a`

(6) Length of latus rectum of parabola is `|4a |`.

(7) Coordinates of ends of latus rectum of parabola are given by

`Y=a` , `X=+-2a`

i.e., `y-k=a` , `X-h =+-2a`

`:.` Ends of latus ractum are `(h+-2a,k+a)`

(8) Parametric equation is `x=h+2at` and `y=k+at^2`

Let `P(x, y)` is any point on the parabola then equation of parabola `y^2 =4ax` is consider as

`(PM)^2 =4a(PN)`

i.e. `(text(The distance of P from its axis))^2 = (text(latus-rectum)) xx`
`quadquadquadquadquadquad (text(The distance of P from the tangent at its vertex))`

where `P` is any point on the parabola.