Conic Sections Calculator
Type any second-degree equation in x and y. It identifies the curve, completes the square, and gives the centre, vertices, foci, asymptotes and eccentricity exactly — then draws it.
(x − 1)²/9 + (y + 2)²/4 = 1
Properties
| Centre | (1, −2) |
|---|---|
| Vertex 1 | (4, −2) |
| Vertex 2 | (−2, −2) |
| Co-vertex 1 | (1, 0) |
| Co-vertex 2 | (1, −4) |
| Focus 1 | (1 + √5, −2)(≈ 3.236, −2) |
| Focus 2 | (1 − √5, −2)(≈ −1.236, −2) |
| a (semi-major) | 3 |
| b (semi-minor) | 2 |
| c (centre to focus) | √5≈ 2.23607 |
| Eccentricity | √5/3≈ 0.745356 |
| Latus rectum | 8/3 |
| Directrix 1 | x = (5 + 9√5)/5 |
| Directrix 2 | x = (5 − 9√5)/5 |
How it was identified
- Compute the discriminant B² − 4ACB² − 4AC = (0)² − 4(4)(9) = −144. Negative, so the curve is an ellipse or a circle — or, if it turns out to be degenerate, a single point or nothing at all.
- Check for a rotationB = 0, so the axes of the conic are parallel to the coordinate axes and no rotation is needed.
- Complete the square in x and in y4(x − 1)² + 9(y + 2)² = 36
- Divide through to reach the standard form(x − 1)²/9 + (y + 2)²/4 = 1
One tool, not four
Most sites split this into a parabola calculator, a circle calculator, an ellipse calculator and a hyperbola calculator. That arrangement asks you to know the answer before you can ask the question, which is backwards when identifying the curve is most of the exercise.
It is also unnecessary. Which conic you have is decided by two numbers computed straight from the coefficients, before anything is drawn or any square is completed:
- B² − 4AC, the discriminant. Negative gives an ellipse or circle, zero a parabola, positive a hyperbola.
- The 3×3 determinant, which is zero exactly when the conic is degenerate — when it collapses to a point, a pair of lines, or nothing at all.
Both are shown on every result, so the classification is something you can check rather than something you have to accept.
The cases most calculators get wrong
A general second-degree equation does not always describe a curve, and the exceptions are a standard exam topic precisely because they catch people out.
| Equation | What it actually is | What a careless tool says |
|---|---|---|
| x² + y² = 0 | A single point, the origin | Circle, radius 0 |
| x² − y² = 0 | Two lines crossing at the origin | Hyperbola with a = b = 0 |
| x² + y² + 1 = 0 | No real points at all | Circle with an imaginary radius |
| x² − 4 = 0 | Two parallel lines, x = ±2 | Parabola |
| x² = 0 | The single line x = 0 | Parabola |
Each of these is detected and named here. The reason a floating-point tool struggles is that the test is “is this exactly zero”, and in floating point that question cannot be answered — the constant that should vanish comes out at 10−17 instead, which is the difference between a point and a very small circle decided by rounding noise. Everything here is computed in exact fractions, so there is no guess to make.
Completing the square
The general form tells you which conic you have. The standard form tells you where it is and how big, and getting from one to the other is completing the square in each variable:
4x² + 9y² − 8x + 36y + 4 = 0
4(x − 1)² + 9(y + 2)² = 36
(x − 1)²/9 + (y + 2)²/4 = 1
From the last line everything else follows by reading: the centre is (1, −2), a² = 9 and b² = 4, so a = 3 and b = 2, and c² = a² − b² = 5 puts the foci at (1 ± √5, −2).
Note which denominator is a². For an ellipse it is always the larger one, and it belongs to whichever variable it sits under — that is what decides whether the major axis is horizontal or vertical, and it is the single most common place to go wrong. For a hyperbola the rule is different: a² belongs to the positive term, whichever it is, and c² = a² + b² rather than a difference.
What each property means
- Foci
- The two points that define the curve. An ellipse is the set of points whose distances to the two foci add to a constant; a hyperbola is the set where the difference is constant. This is why a whisper gallery works, and why a satellite dish has a receiver at one point.
- Eccentricity
- c/a — how far from circular. Exactly 0 for a circle, between 0 and 1 for an ellipse, exactly 1 for a parabola, greater than 1 for a hyperbola. One number that classifies the whole family.
- Directrix
- A line paired with each focus. Every point on the conic has the same ratio of its distance to the focus and its distance to the directrix, and that ratio is the eccentricity. For a parabola the two distances are equal, which is the definition most courses start from.
- Asymptotes
- The two lines a hyperbola approaches but never reaches. Their slopes are ±b/a for a horizontal transverse axis, and drawing them first is the fastest way to sketch the curve by hand.
- Latus rectum
- The chord through a focus perpendicular to the major axis. Its length, 2b²/a, gives you the width of the curve at the focus and a second point to sketch through.
Rotated conics, and what is honestly possible
When the equation has an xy term, the conic’s axes are not parallel to the coordinate axes. The classification still works — the discriminant and determinant are polynomials in the coefficients, so they stay exact — and the rotation angle follows from cot 2θ = (A − C)/B.
What does not survive is exactness in the original coordinates. Removing the xy term requires rotating by an angle that is an arctangent, and the centre, vertices and foci in the rotated frame are irrational in a way no surd captures. So this tool classifies a rotated conic, gives the angle, and then says plainly that it will not report vertices. Printing decimals from a frame the student did not ask about would look more helpful and be less true.
Four mistakes this will catch
- Assuming the larger denominator goes with x. For an ellipse, a² is the larger denominator wherever it sits — and it decides the orientation. Reading it off the wrong variable puts the vertices on the wrong axis.
- Using c² = a² − b² for a hyperbola. That is the ellipse relation. A hyperbola has c² = a² + b², so its foci are further out than its vertices, not closer in.
- Dropping a sign when completing the square. Subtracting the added constant from the correct side is where most of the arithmetic goes wrong, and the standard form shown here is the check.
- Calling every second-degree equation a conic. Some of them are points, lines or nothing. The determinant says which, and it is shown on every result.
What it does not do
- Vertices and foci of a rotated conic, for the reason given above.
- Polar forms, or conics given by a focus and directrix rather than an equation.
- Three-dimensional quadric surfaces.
- Tangent lines, or intersections between two conics.
- Complex coefficients.
Questions
How does it know which conic I typed without my telling it?
From the discriminant B² − 4AC. Negative means an ellipse or a circle, zero means a parabola, positive means a hyperbola — and that is a fact about the coefficients, not about the shape of the graph, so it can be computed before anything is drawn. The 3×3 determinant then decides whether the conic is the real thing or one of its degenerate forms. Both are exact rational computations.
Why does x² + y² = 0 come out as a point rather than a circle of radius zero?
Because that is what it is. Two squares with the same sign can only sum to zero when both are zero, so exactly one point in the plane satisfies the equation. Calling it a circle of radius zero is the kind of answer that sounds right and is not: a circle is a curve, and this has no curve in it. The same applies to x² − y² = 0, which is a pair of crossing lines rather than a hyperbola, and to x² + y² + 1 = 0, which nothing satisfies at all.
Why is the focus given as (1 + √5, −2) instead of (3.236, −2)?
Because the surd is the answer and the decimal is a photograph of it. A course that sets a conic with a non-square c² is asking for the exact form, and a marker will want to see it. The decimal is shown underneath, so you have both — but the exact value is the one that goes on the page.
What does the eccentricity actually tell me?
How far the conic is from being a circle. Eccentricity is zero for a circle, between 0 and 1 for an ellipse, exactly 1 for a parabola and greater than 1 for a hyperbola — so it is a single number that classifies the whole family. An ellipse with eccentricity 0.1 is nearly round; one with 0.99 is a long thin cigar. Earth’s orbit is about 0.017.
Why will it not give me the vertices of xy = 1?
Because that conic is rotated, and its vertices are not expressible as exact fractions or surds in the coordinates you typed. It is a hyperbola at 45°, which the tool says, and the rotation angle is exact — but the centre, vertices and foci would need working in a rotated frame, and reporting decimals from that frame as though they belonged to the original one would be misleading. Saying what cannot be done is more useful than a plausible number.
Can I type it in any form?
Any second-degree polynomial in x and y, with or without an equals sign, expanded or in brackets. (x − 1)² + (y + 2)² = 9 and x² + y² − 2x + 4y − 4 = 0 are the same input as far as this is concerned. Fractions and decimal coefficients stay exact — 0.25x² is read as a quarter of x², not as the nearest floating-point number to it.