Introduction
A conic is the locus of a point whose distance from a fixed point bears a constant ratio to its distance from a fixed line. The fixed point is the focus S and the fixed line is the directrix l
The constant ratio is called the eccentricity denoted by ‘e’
- if 0 < e < 1, the conic is an ellipse.
- e = 1, conic is a parabola
- e > 1, conic is a hyperbola.
Important Terms Related to Parabola
- Axis: A line perpendicular to thee directrix and passes through the focus.
- Vertex: The intersection point of the conic and axis.
- Centre: The point which bisects every chord of the conic passing through it.
- Focal Chord: Any chord passing through the focus.
- Latus Rectum: A double ordinate passing through the focus of the parabola.
Standard Form of a Parabola and Related Terms
| Equation of Parabola | y2 = 4ax |
 |
|
| Vertex | (0, 0) |
| Focus | (a, 0) |
| Equation of axis | y = 0 |
| Equation of directrix | x + a = 0 |
| Eccentricity | e = 1 |
| Extremities of Latus Rectum | (a, 2a) and (a, -2a) |
| Length of Latus Rectum | 4a |
| Parametric equation | x = at2 and y = 2at |
| Equation of Latus rectum | x – a = 0 |
| Equation of Parabola | y2 = – 4ax |
 |
|
| Vertex | (0, 0) |
| Focus | (- a, 0) |
| Equation of axis | y = 0 |
| Equation of directrix | x – a = 0 |
| Eccentricity | e = 1 |
| Extremities of Latus Rectum | (-a, 2a) and (-a, -2a) |
| Length of Latus Rectum | 4a |
| Parametric equation | x = – at2 and y = 2at |
| Equation of Latus rectum | x + a = 0 |
| Equation of Parabola | x2 = 4ay |
 |
|
| Vertex | (0, 0) |
| Focus | (0, a) |
| Equation of axis | x = 0 |
| Equation of directrix | y + a = 0 |
| Eccentricity | e = 1 |
| Extremities of Latus Rectum | (2a, a) and (-2a, a) |
| Length of Latus Rectum | 4a |
| Parametric equation | x = 2at and y = at2 |
| Equation of Latus rectum | y – a = 0 |
| Equation of Parabola | x2 = 4ay |
 |
|
| Vertex | (0, 0) |
| Focus | (0, – a) |
| Equation of axis | x = 0 |
| Equation of directrix | y – a = 0 |
| Eccentricity | e = 1 |
| Extremities of Latus Rectum | (2a, – a) and (-2a, – a) |
| Length of Latus Rectum | 4a |
| Parametric equation | x = 2at and y = – at2 |
| Equation of Latus rectum | y + a = 0 |
| Equation of Parabola | (y – β)2 = 4a (x – α) |
| Vertex | (α, β) |
| Focus | (a + α, β) |
| Equation of axis | Parallel to x – axis |
| Equation of directrix | x + a = α |
| Eccentricity | 1 |
| Extremities of Latus Rectum | (α + a, β + 2a) and (α + a, β – 2a) |
| Length of Latus Rectum | 4a |
| Equation of Parabola | (x – α)2 = 4a (y – β) |
| Vertex | (α, β) |
| Focus | (α, a + β) |
| Equation of axis | Parallel to y – axis |
| Equation of directrix | y + a = β |
| Eccentricity | 1 |
| Extremities of Latus Rectum | (α + 2a, β + a) and (α – 2a, β + a) |
| Length of Latus Rectum | 4a |
Position of a Point
The point (x1, y1) lies outside, on or inside the parabola y2 = 4ax according as y1 – 4ax1 >, =, < 0.
| y1 – 4ax1 > 0 | outside the parabola |
| y1 – 4ax1 = 0 | on the parabola |
| y1 – 4ax1 < 0 | inside the parabola |
