A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). To turn this definition into a clean equation, NCERT places the parabola in the simplest position: vertex at the origin with its axis along a coordinate axis. The equations you get are called the standard equations.
The four standard forms
Depending on which way the parabola opens, there are four standard equations. In each, a>0.
Equation
Opens
Focus
Directrix
y2=4ax
right
(a,0)
x=−a
y2=−4ax
left
(−a,0)
x=a
x2=4ay
up
(0,a)
y=−a
x2=−4ay
down
(0,−a)
y=a
For all four the vertex is at the origin (0,0) and the axis of the parabola is a coordinate axis.
Where y2=4ax comes from
Take the focus at F(a,0) and the directrix as the line x=−a. For a point P(x,y) on the parabola, its distance to the focus equals its distance to the directrix:
(x−a)2+y2=x+a.
Squaring both sides:
(x−a)2+y2=(x+a)2,
and expanding gives y2=4ax. The other three forms follow by turning the focus in a different direction.
Latus rectum
The latus rectum is the chord through the focus, perpendicular to the axis, with both ends on the parabola. For every standard parabola its length is 4a — the very same 4a that appears in the equation, which makes it quick to read off.
Worked example
For the parabola y2=12x, compare with y2=4ax: here 4a=12, so a=3.
A point on the parabola is equidistant from the focus and the directrix. Setting (distance to focus)=(perp. distance to directrix) and squaring gives 4x2+y2+4xy+4x+32y+16=0.
A parabola is the locus of points P(x,y) equidistant from a fixed point (focus) and a fixed line (directrix). With focus F(−1,−2) and directrix x−2y+3=0:
1. Distance to the focus
dF=(x+1)2+(y+2)2
2. Perpendicular distance to the directrix (using a2+b2∣ax0+by0+c∣ with a=1,b=−2,c=3)