Definition of Parabola
If S and L are a point and a fixed straight line (not passing through S) in a plane, respectively, and a point P in that plane is moving in such a way that the point P is always equidistant from the point S and the straight line L in all locations, then the locus of point P is called Parabola.
The fixed point S is called the focus of the parabola and the fixed straight line L is called the directrix of the parabola.
The perpendicular from focus S to the directrix L is called the axis of the parabola.
The point where the parabola intersects with its axis is called the vertex.
Any chord of the parabola passing through the focus is called the focal chord.
The length of all the chords perpendicular to the axis of the parabola is called double ordinate.
The double ordinate passing through the focus of the parabola is called the latus rectum.
In the above figure straight line L – directrix, point S – focus, straight-line AS – axis, point A – vertex, PN = ordinate of point P, PNQ = double ordinate of the point P, AN = abscissa of the point P, L1L2 = latus rectum, PSR = chord passing through focus.
The standard equation of a parabola is {{y}^{2}}=4ax where ‘a’ is the between focus and the vertex of the parabola and x-axis is the axis of the parabola.
Some Formulae on Parabola
1. In the following a = is the between focus and the vertex of the parabola
| Equation of Parabola | Coordinate of the Vertex | Axis | Coordinate of the Focus | Length of the Latus Rectum | Equation of the Directrix |
| \[{{y}^{2}}=4ax\] | (0, 0) | Positive x-axis | (a, 0) | 4a units | x + a = 0 |
| \[{{y}^{2}}=-4ax\] | (0, 0) | Negative x-axis | (–a, 0) | 4a units | x – a = 0 |
| \[{{x}^{2}}=4ay\] | (0, 0) | Positive y-axis | (0, a) | 4a units | y + a = 0 |
| \[{{x}^{2}}=-4ay\] | (0, 0) | Negative y-axis | (0, –a) | 4a units | y – a = 0 |
| \[{{\left( y-\beta \right)}^{2}}=4a\left( x-\alpha \right)\] | (α, β) | Parallel to x-axis | (a + α, β) | 4a units | x + a = α |
| \[{{\left( x-\alpha \right)}^{2}}=4a\left( y-\beta \right)\] | (α, β) | Parallel to y-axis | (α, a + β) | 4a units | y + a = β |
2. x=a{{y}^{2}}+by+c\,\,\left( a\ne 0 \right) represents the equation of a parabola with its axis parallel to x-axis.
3. y=p{{x}^{2}}+qx+r\,\,\left( p\ne 0 \right) represents the equation of a parabola with its axis parallel to y-axis.
4. P(x1, y1) point will be inside, outside or on the parabola {{y}^{2}}=4ax if the value of \left( y_{1}^{2}=4a{{x}_{1}} \right) > or < or = 0 respectively.
5. We can take any point P on {{y}^{2}}=4ax as \left( a{{t}^{2}},2at \right).
6. The parametric equation of {{y}^{2}}=4ax is x=a{{t}^{2}},y=2at. Here t is the parameter.
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