The length of the latus rectum (LR) of a parabola is determined by the absolute value of the coefficient ‘a’ in its equation. For a vertical parabola (opens upward or downward) with the equation y = ax^2, the LR is 4/|a|. For a horizontal parabola (opens right or left) with x = ay^2, the LR is 4/|a|.
Latus Rectum Calculator
| Type of Parabola | Equation | Length of Latus Rectum (LR) |
|---|---|---|
| Vertical Parabola (opens upward) | y = ax^2 | LR = 4/ |
| Vertical Parabola (opens downward) | y = -ax^2 | LR = 4/ |
| Horizontal Parabola (opens to right) | x = ay^2 | LR = 4/ |
| Horizontal Parabola (opens to left) | x = -ay^2 | LR = 4/ |
FAQs
The directrix of a parabola is a fixed line that is used in the definition of the parabola. The directrix is located at an equal distance from the vertex of the parabola as the focus. The focus is a point on the parabola, and the directrix is a straight line.
The formula for finding the directrix of a parabola depends on whether the parabola is vertical or horizontal and its orientation. Here are the formulas for the directrix of a parabola:
- Vertical Parabola with Vertex at (h, k):
- For a parabola that opens upward: y = k – p
- For a parabola that opens downward: y = k + p
- Horizontal Parabola with Vertex at (h, k):
- For a parabola that opens to the right: x = h – p
- For a parabola that opens to the left: x = h + p
In these formulas, (h, k) represents the vertex of the parabola, and ‘p’ is the distance between the vertex and the focus.
To find the focus of a parabola given the vertex and directrix, you would use the same distance ‘p’ and the opposite direction. If the directrix is above the vertex, then the focus is below, and vice versa. Similarly, if the directrix is to the left of the vertex, then the focus is to the right, and vice versa.
Yes, a parabola always has a directrix. The directrix is a fundamental geometric element of a parabola.
The formula for the focus of a parabola is as follows:
- For a parabola that opens upward or downward with vertex (h, k):
- Focus = (h, k + p) for upward opening parabola or (h, k – p) for downward opening parabola.
- For a parabola that opens to the right or left with vertex (h, k):
- Focus = (h + p, k) for right opening parabola or (h – p, k) for left opening parabola.
The formula for a parabola in general form (ax^2 + bx + c = 0) or vertex form (a(x-h)^2 + k) depends on the specific equation of the parabola, and it is used to describe the overall shape and position of the parabola. The directrix and focus are derived from this equation based on the form and orientation of the parabola.
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