Parabola Focus and Directrix Calculator – Find Every Key Property
A parabola is the set of all points that are equally distant from a fixed point, called the focus, and a fixed line, called the directrix. This geometric definition is the foundation of satellite dishes, headlight reflectors, suspension bridge cables, and projectile motion in physics. The Parabola Focus and Directrix Calculator extracts every important property of a parabola — vertex, focus, directrix, axis of symmetry, focal length, and latus rectum — directly from its equation.
Vertex Form vs. Standard Conic Form
The calculator accepts a parabola in either of two common notations:
- Vertex form:
y = a(x - h)² + kfor a vertical parabola, orx = a(y - k)² + hfor a horizontal one. - Standard conic form:
(x - h)² = 4p(y - k)for vertical, or(y - k)² = 4p(x - h)for horizontal.
Both forms describe the same curve — the tool converts between them automatically and shows both equations in the results.
Core Formulas
For a vertical parabola y = a(x - h)² + k, the focal length is p = 1 / (4a). From there:
- Focus:
(h, k + p) - Directrix:
y = k - p - Axis of symmetry:
x = h - Latus rectum length:
|4p| = |1/a|
The parabola opens up when a > 0 and down when a < 0. For a horizontal parabola x = a(y - k)² + h, the same p is used, but the focus becomes (h + p, k), the directrix is x = h - p, and the parabola opens right when a > 0 or left when a < 0.
Worked Example
Consider y = 2(x - 3)² - 1, so a = 2, h = 3, k = -1:
p = 1 / (4 × 2) = 0.125Focus = (3, -1 + 0.125) = (3, -0.875)Directrix: y = -1 - 0.125 = -1.125Axis of symmetry: x = 3Latus rectum length = |1/2| = 0.5
Reverse Mode: From Focus and Directrix to Equation
Sometimes the focus and directrix are known and the equation must be derived — for example, when designing a parabolic reflector to a target focal point. Given a focus (fx, fy) and a horizontal directrix y = d, the vertex is the midpoint between them: k = (fy + d) / 2 and h = fx. The focal length follows as p = fy - k, and the coefficient is a = 1 / (4p).
The Latus Rectum
The latus rectum is the chord that passes through the focus, perpendicular to the axis of symmetry, with both endpoints on the parabola. Its length, |4p|, gives a useful measure of how "wide" the parabola is near the focus — a larger latus rectum means a wider, more open curve, while a smaller one means a narrower, sharper curve.
Practical Applications
- Antenna and satellite dish design — signals arriving parallel to the axis reflect toward the focus, so engineers must know the exact focal point to place the receiver.
- Reflector and headlight design — a light source placed at the focus produces a beam of parallel rays after reflecting off a parabolic mirror.
- Physics and projectile motion — the trajectory of a projectile under constant gravity traces a parabola, and its vertex and axis of symmetry describe the peak and symmetry of the flight path.
- Pre-calculus and analytic geometry education — verifying homework and building intuition for how a, h, and k affect the shape and position of a parabola.
Tips for Accurate Results
a (or the 4p value in standard form) can never be zero — a zero coefficient describes a straight line, not a parabola.- Double-check whether your parabola is vertical (opens up/down) or horizontal (opens left/right) before reading the focus and directrix formulas.
- Use a higher decimal precision when the result feeds into further engineering calculations to minimize rounding errors.
- In reverse mode, remember the focus can never lie exactly on the directrix — that configuration has no valid parabola.