Calculator
Choose an equation style. Then enter its matching values. The answer appears above this form after submission.
Example Data Table
| Equation type | Input values | Focus result | Main formula |
|---|---|---|---|
| y = ax² + bx + c | a = 1, b = -4, c = 3 | (2, -0.75) | p = 1 / 4a |
| Vertex parabola | h = 1, k = 2, p = 3 | (1, 5) | (x - h)² = 4p(y - k) |
| Ellipse | h = 0, k = 0, a = 5, b = 3 | (-4, 0), (4, 0) | c² = a² - b² |
| Hyperbola | h = 0, k = 0, a = 4, b = 3 | (-5, 0), (5, 0) | c² = a² + b² |
Formula Used
Vertical parabola: (x - h)² = 4p(y - k). Focus is (h, k + p). Directrix is y = k - p.
Horizontal parabola: (y - k)² = 4p(x - h). Focus is (h + p, k). Directrix is x = h - p.
Quadratic parabola: For y = ax² + bx + c, use h = -b / 2a, k = c - b² / 4a, and p = 1 / 4a.
Ellipse: Use c² = a² - b². Foci sit c units from the center along the major axis.
Hyperbola: Use c² = a² + b². Foci sit c units from the center along the transverse axis.
How to Use This Calculator
- Select the equation type that matches your conic.
- Enter coefficients, vertex values, center values, or axis lengths.
- Choose the correct horizontal or vertical direction when needed.
- Press the calculate button.
- Read the focus coordinates, vertex, directrix, and graph.
- Use the CSV or PDF button to save the result.
Focus of an Equation Guide
What the Focus Means
The focus is a special point in a conic section. It controls the shape and position of the curve. A parabola has one focus. An ellipse has two foci. A hyperbola also has two foci. These points help describe reflection, distance, and symmetry. They are used in geometry, optics, astronomy, and engineering.
Why Equation Form Matters
Different equations show the focus in different ways. Vertex form is the easiest for a parabola. It gives the vertex and the p value directly. Standard ellipse form gives the center and axis lengths. Hyperbola form gives the center and transverse direction. A general equation needs completing the square before the focus can be found.
Parabola Focus
For a vertical parabola, the focus sits above or below the vertex. The p value decides the distance. If p is positive, the curve opens upward. If p is negative, it opens downward. For a horizontal parabola, the focus sits right or left of the vertex. The same p value controls that shift.
Ellipse Focus
An ellipse has two focus points inside the curve. They lie on the major axis. The distance from the center is called c. It is found from c squared equals a squared minus b squared. A larger gap between a and b moves the foci farther from the center.
Hyperbola Focus
A hyperbola has two branches and two focus points. The foci lie outside the center. Their distance uses c squared equals a squared plus b squared. This makes c greater than a. The transverse axis decides whether the foci move left and right or up and down.
Practical Use
This calculator saves time when checking algebra. It also helps compare several conic forms. The graph gives a visual check. The export buttons help record your answer. Use exact input values when possible. Then round the final coordinates only when your class or project requires it.
FAQs
1. What is the focus of a parabola?
The focus is a fixed point used to define the parabola. Every point on the parabola is equally distant from the focus and the directrix.
2. Can this calculator handle sideways parabolas?
Yes. Choose the sideways quadratic or horizontal vertex form. The calculator then uses the horizontal focus formula.
3. What does p mean in a parabola?
The p value is the directed distance from the vertex to the focus. Its sign controls the opening direction.
4. Why does an ellipse have two foci?
An ellipse is based on the sum of distances from two fixed points. Those fixed points are the two foci.
5. Why is c different for hyperbolas?
For hyperbolas, c squared equals a squared plus b squared. This places the foci outside the center region.
6. What if my equation is in general form?
Use the general equation options. The calculator completes the square and changes the equation into vertex form.
7. Can I export my answer?
Yes. After calculation, use the CSV button for spreadsheet data or the PDF button for a printable summary.
8. Does the graph update after calculation?
Yes. The graph shows the conic, vertex or center, and focus points based on the submitted values.