As discussed earlier, there are four standard forms of a parabola. In short, they represent
- A right handed parabola: `y^2 = 4ax`
- A left handed parabola: `y^2 = -4ax`
- An up facing parabola: `x^2 = 4ay`
- A down facing parabola: `x^2 = -4ay`
In each of these equations, `x` and `y` are variables and `a` is any number.
Now as you may know, a parabola has some characteristic properties, namely the
vertex,
focus,
directrix,
axis,
latus rectum, etc. Each of the four standard forms have their own set of properties. These are discussed below:
`y^2 = 4ax` or Right Handed Parabola
- Focus: (a, 0)
- Directrix: x = -a
- Vertex: (0, 0)
- Axis: y = 0 (that is, the x-axis)
- Length of latus rectum: 4a
For example, for the equation `y^2 = 2x`, comparing it with standard form `y^2 = 4ax`, we get a 4a = 2, that is, a = 1/2. Therefore the focus, directrix, etc. for `y^2 = 2x` are as follows:
- Focus: (1/2, 0)
- Directrix: x = -1/2
- Vertex: (0, 0)
- Axis: y = 0 (that is, the x-axis)
- Length of latus rectum: 4(1/2) = 1/2
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| `y^2 = 2x` and its characteristics |
`y^2 = -4ax` or Left Handed Parabola
- Focus: (-a, 0)
- Directrix: (a, 0)
- Vertex: (0, 0)
- Axis: y = 0 (that is, the x-axis)
- Length of latus rectum: 4a
For example, for the equation `y^2 = -2x`, comparing it with standard form `y^2 = -4ax`, we get a 4a = 2, that is, a = 1/2. Therefore the focus, directrix, etc. for `y^2 = -2x` are as follows:
- Focus: (-1/2, 0)
- Directrix: x = 1/2
- Vertex: (0, 0)
- Axis: y = 0 (that is, the x-axis)
- Length of latus rectum: 4(1/2) = 1/2
`x^2 = 4ay` or Up Facing Parabola
- Focus: (0, a)
- Directrix: (0, -a)
- Vertex: (0, 0)
- Axis: x = 0 (that is, the y-axis)
- Length of latus rectum: 4a
For example, for the equation `x^2 = 2y`, comparing it with standard form `x^2 = 4ay`, we get a 4a = 2, that is, a = 1/2. Therefore the focus, directrix, etc. for `x^2 = 2y` are as follows:
- Focus: (0, 1/2)
- Directrix: y = -1/2
- Vertex: (0, 0)
- Axis: x = 0 (that is, the x-axis)
- Length of latus rectum: 4(1/2) = 1/2
 |
| `x^2 = 2y` and its characteristics |
`x^2 = -4ay` or Down Facing Parabola
- Focus: (0, -a)
- Directrix: (0, a)
- Vertex: (0, 0)
- Axis: x = 0 (that is, the y-axis)
- Length of latus rectum: 4a
For example, for the equation `x^2 = -2y`, comparing it with standard form `x^2 = -4ay`, we get a 4a = 2, that is, a = 1/2. Therefore the focus, directrix, etc. for `x^2 = -2y` are as follows:
- Focus: (0, -1/2)
- Directrix: y = +1/2
- Vertex: (0, 0)
- Axis: x = 0 (that is, the x-axis)
- Length of latus rectum: 4(1/2) = 1/2
 |
| `x^2 = -2y` and its characteristics |
