Parábola — Foco
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
foco da parábola
About this calculator
This calculator determines the focus of a parabola from its equation in canonical form (x - h)² = 4p(y - k) or (y - k)² = 4p(x - h). The focus is a fixed point that, together with the directrix, defines the parabola as the set of points equidistant from the focus and directrix. The calculator identifies the value of p, the distance from the vertex to the focus, and then calculates the focus coordinates based on the vertex (h, k) and the parabola's orientation.
The operation is simple: enter the parabola equation in the accepted formats. The tool extracts coefficients and identifies whether the parabola is vertical (opens up or down) or horizontal (opens left or right). Then it calculates p = 1/(4a), where a is the coefficient of the quadratic term after rewriting the equation in standard form. The focus is given by (h, k + p) for vertical parabolas or (h + p, k) for horizontal ones.
Use this calculator when you need to quickly find the focus of a parabola in analytic geometry problems, physics (such as parabolic reflectors), or engineering. For example, when designing a parabolic antenna, the focus is the point where the signal should be captured. It is also useful for students to check exercises on parabola properties.
Caution: ensure the equation is in correct canonical form. If the equation is in general form (ax² + bx + c = y), you must complete the square before using. The calculator assumes the equation is already simplified. Also, check that the coefficient of the quadratic term is nonzero; otherwise, the equation does not represent a parabola.
Frequently asked questions
What is the focus of a parabola?
The focus is a fixed point used to define the parabola. The parabola is the set of points equidistant from the focus and a line called the directrix.
How do I know if the parabola opens up, down, left, or right?
In the form (x - h)² = 4p(y - k), if p > 0 it opens up; if p < 0 it opens down. In the form (y - k)² = 4p(x - h), if p > 0 it opens right; if p < 0 it opens left.
Can I use the calculator with the equation in general form y = ax² + bx + c?
Not directly. The calculator expects the canonical form. You must first complete the square to get (x - h)² = 4p(y - k) and then enter the values of h, k, and p.
What is the relationship between the focus and the directrix?
The focus and directrix are at a distance p from the vertex. The focus lies on the opposite side of the directrix from the vertex, and the parabola is symmetric about the axis through the focus and vertex.
What happens if p is zero?
If p = 0, the equation does not represent a parabola because the distance between focus and directrix would be zero. In that case, the equation reduces to a line.