Parabola is a conic section. It is defined as the set of all points whose distance from a fixed point, called the focus, is equal to the distance from a fixed line, called the directrix.
The eccentricity, e of a parabola is equal to 1.
| Terms | |
|---|---|
| Focus (F) | \(a\) distance from the vertex. |
| Vertex (v) | \(h,k\) |
| Latus Rectum (LR) | \(LR~=~4a\) A line segment passing through the focus and parallel to the directrix. |
| \(a\) | distance between the vertex and the focus or the distance between the directix and the vertex. |
General Equation of Parabola
\( Ax^2 + Dx + Ey + F = 0\) or \(Cy^2 +Dx + Ey + F = 0\)
Where A, B, C, D E and F are constants.
Standard Equations of Parabola
| Vertex at \( (0,0) \) | |
|---|---|
| \(y^2 = 4ax\) | Opens Upwards |
| \(y^2 = -4ax\) | Opens Downwards |
| \(x^2 = 4ax\) | Opens to the right |
| \(x^2 = -4ax\) | Opens to the left |
| Vertex at \( (h,k) \) | |
| \( (y-h)^2 = 4a(x-k)\) | Opens Upwards |
| \((y-h)^2 = -4a(x-k)\) | Opens Downwards |
| \( (x-k)^2 = 4a(y-h)\) | Opens to the right |
| \((x-k)^2 = -4a(y-h)\) | Opens to the left |
How to solve \(h,k\) and \(a \) if the given equation is in its general form.
- Transform to standard form and get the \(h,k\) values.
- If the general form is: \( Ax^2 + Dx + Ey + F = 0\)
- \(h = \frac{-D}{2A}\)
- \(k = \frac{D^2-4AF}{4AE}\)
- \(a = \frac{-E}{4A}\)
- If the general form is: \(Cy^2 +Dx + Ey + F = 0\)
- \(h = \frac{E^2-4CF}{4CD}\)
- \(k = \frac{-E}{2C}\)
- \(a = \frac{-D}{4C}\)