Coordinate geometry
Conic sections from the equation
Enter the six coefficients of Ax² + Bxy + Cy² + Dx + Ey + F = 0: the calculator identifies which conic it is and works out the centre or vertex, foci, axes, eccentricity, asymptotes and canonical form, drawing it to scale.
Recognising a conic from its equation
Every second-degree equation in two unknowns describes a conic, and a single number says which: the discriminant B² − 4AC. Negative and the curve is closed — an ellipse, or a circle when A = C and B = 0; zero and it is a parabola; positive and it is a hyperbola. This is the first step of any conic-sections exercise, and it is done in your head before any other formula is touched.
The second number to look at is the determinant of the 3×3 matrix of coefficients. When it vanishes the polynomial factorises and the conic degenerates: in place of an ellipse only the centre is left, in place of a hyperbola the two lines that were its asymptotes, in place of a parabola two parallel lines. These are legitimate cases to be recognised, not mistaken for flattened curves.
Once the species is known the rest follows from the canonical form, the equation written with the conic's own axes. For a central conic the centre comes first — the point where both partial derivatives vanish — and the axes are moved there; if the xy term is present they are turned as well, through an angle given by tan 2θ = B/(A − C). On those axes the equation has only the squares, and the semi-axes, foci and eccentricity can be read straight off.
Common mistakes
- Rearranging into the required form and forgetting a sign: y = x² − 4x + 3 becomes x² − 4x − y + 3 = 0, with E = −1 rather than +1.
- Taking the semi-major axis to be the one multiplying x²: in x²/9 + y²/25 = 1 the semi-major axis is 5 and lies on the y-axis, not 3.
- Looking for a parabola's foci on the wrong axis: a parabola has one focus, on its own axis of symmetry, and the directrix sits on the far side of the vertex at the same distance.
- Assuming a hyperbola's eccentricity lies between 0 and 1: it is always greater than 1, and it is the ellipse that has e < 1.
Frequently asked questions
How can you tell an ellipse, a parabola and a hyperbola apart at a glance?
By the sign of B² − 4AC. Negative: ellipse or circle. Zero: parabola. Positive: hyperbola. It needs neither the centre nor any rotation, which is why it is always the first step.
What does it mean for a conic to be degenerate?
That the second-degree polynomial factorises, so the set of points satisfying it is no longer a curve but a point, a pair of lines, or nothing at all. It shows up in the determinant of the 3×3 matrix of coefficients: when that is zero, the conic is degenerate.
What is the xy term for?
It tilts the conic relative to the coordinate axes. Without it the curve's axes are parallel to the plane's; with it they make an angle θ given by tan 2θ = B/(A − C). The simplest example is xy = 1, a rectangular hyperbola turned through 45°.
Why is a circle's eccentricity zero?
Because eccentricity measures how far an ellipse departs from being circular: it is c/a, where c is the distance from the centre to a focus. In a circle both foci sit at the centre, so c = 0 and e = 0. For an ellipse 0 < e < 1, for a parabola e = 1, for a hyperbola e > 1.
Do ellipses have asymptotes too?
No: asymptotes are lines a curve approaches indefinitely without meeting, and only the hyperbola has them. An ellipse is bounded, fitting entirely inside a rectangle of sides 2a and 2b, and approaches no line at all.
How this calculation works
From the general form Ax² + Bxy + Cy² + Dx + Ey + F = 0 build the full matrix M, with rows (A, B/2, D/2), (B/2, C, E/2), (D/2, E/2, F), and its submatrix A₃₃ of the first two rows and columns. The sign of det A₃₃, that is of −(B² − 4AC)/4, gives the species; det M = 0 gives degeneracy. For a central conic the centre solves 2Ax + By + D = 0 and Bx + 2Cy + E = 0, the eigenvalues λ₁ and λ₂ of A₃₃ are the coefficients of the squares in the canonical form λ₁X² + λ₂Y² + det M/det A₃₃ = 0, and the squared semi-axes are −(det M/det A₃₃)/λ. The rotation angle is half the argument of (A − C, B). For a parabola one eigenvalue is zero: rotate, complete the square on the other variable and reach X² = 4pY, with p the vertex-to-focus distance.
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