KEVOS® MAKEOS · Mathematics
Conic Section from Its General Equation (Parabola, Ellipse, Hyperbola: Centre, Vertices, Foci, Directrix)
Given the equation Ax² + Cy² + Dx + Ey + F = 0 of a curve, is it a parabola, an ellipse (or circle) or a hyperbola, and where are its vertex or centre, vertices, foci, directrix, axes and asymptotes?
Identifies the conic section of a general second-degree equation without an xy term and puts it into the standard form by completing the squares, as in the handbook examples: for a parabola (y − k)² = 4p(x − h) or (x − h)² = 4p(y − k), giving vertex, focus, directrix, axis and latus rectum; for an ellipse or circle the centre, semi-axes, foci and eccentricity; for a hyperbola (x − h)²/a² − (y − k)²/b² = 1 (or the vertical form) the centre, a, b, c = √(a² + b²), the eccentricity e = c ÷ a, vertices, foci and asymptotes.
Open the calculatorRuns in your browser. No account needed.
How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Coefficient A of x² | Number multiplying x² in the equation Ax² + Cy² + Dx + Ey + F = 0 (enter 0 when there is no x² term). | |
| Coefficient C of y² | Number multiplying y² in the equation (enter 0 when there is no y² term). | |
| Coefficient D of x | Number multiplying x in the equation, with its sign. | |
| Coefficient E of y | Number multiplying y in the equation, with its sign. | |
| Constant F | Constant term of the equation, with its sign. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Coefficient A of x²
- What it is
- Number multiplying x² in the general equation Ax² + Cy² + Dx + Ey + F = 0.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 9 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- With C, decides whether the curve is a parabola, ellipse or hyperbola.
- Get it right
- Enter 0 when the equation has no x² term; keep the sign.
What this means: Coefficient C of y²
- What it is
- Number multiplying y² in the general equation of the curve.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -4 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- With A, decides the kind of curve and the length of its axes.
- Get it right
- Enter 0 when the equation has no y² term; keep the sign.
What this means: Coefficient D of x
- What it is
- Number multiplying x in the general equation, with its sign.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -36 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Shifts the curve along x.
- Get it right
- Move every term to one side so the equation equals zero first.
What this means: Coefficient E of y
- What it is
- Number multiplying y in the general equation, with its sign.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 8 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Shifts the curve along y.
- Get it right
- Move every term to one side so the equation equals zero first.
What this means: Constant F
- What it is
- Constant term of the general equation, with its sign.
- Units and format
- Enter a plain number with no unit. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -4 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the size of an ellipse or hyperbola and the position of a parabola.
- Get it right
- It is the number left with no x or y once the equation equals zero.
Method and formulas
- Parabola (A or C zero): complete the square in the squared variable to reach (y − k)² = 4p(x − h) or (x − h)² = 4p(y − k); focus at p from the vertex along the axis, directrix at −p, latus rectum through the focus with length |4p|.
- Otherwise A(x − h)² + C(y − k)² = G with h = −D ÷ 2A, k = −E ÷ 2C and G = D²/4A + E²/4C − F.
- AC > 0: ellipse with semi-axes √(G ÷ A) along x and √(G ÷ C) along y (a circle when they are equal); c = √(a² − b²) along the major axis, e = c ÷ a.
- AC < 0: hyperbola; its transverse axis is along x when G ÷ A > 0; c² = a² + b², e = c ÷ a, asymptotes through the centre with slopes ± b ÷ a (x axis) or ± a ÷ b (y axis).
- Example: 9x² + (-4)y² + (-36)x + 8y + (-4) = 0 is a hyperbola with vertex or centre (2, 1) and eccentricity 1.8028.
Limitations
- Equations with an xy term (rotated conics) are not covered.
- Equations that reduce to a point, a pair of lines or no real curve are reported as such.
- The handbook text says a hyperbola’s eccentricity is always less than 1; it is always greater than 1, as its own example (e = √13 ÷ 2) shows.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the focus of a parabolic reflector or profile from its equation
- Check a curve equation from a drawing or CAD export
- Work the conic examples in a maths course
Who it is for
- Students
- Designers
- CNC programmers
What you get
- Kind of curve
- Vertex or centre (h, k)
- Parabola: p, focus, directrix, latus rectum
- Ellipse or hyperbola: a, b, c and eccentricity
- Vertices, foci and asymptotes
Interpreting the result
- Eccentricity 0 is a circle, between 0 and 1 an ellipse, 1 a parabola and above 1 a hyperbola.
- Multiplying the whole equation by a number does not change the curve.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 54–55. Topic reference; no source tables or recommended settings are reproduced.
KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.