Conic Sections Explorer

implicit conic curve from Ax² + Bxy + Cy² + Dx + Ey + F = 0

Worked example — Conic Sections Explorer

Equation	0
Classification rule	Δ<0⇒Ellipse,Δ=0⇒Parabola,Δ>0⇒Hyperbola
Classification	Ellipse

These are the values the calculator shows when the page opens. Change any input above to recompute them.

Loading interactive simulation...

discriminant is a geometric phase boundary 🖖

The discriminant Δ = B² - 4AC is the phase boundary for conics: Δ < 0 gives ellipse-family curves, Δ = 0 gives parabola-family curves, and Δ > 0 gives hyperbola-family curves. Linear terms D and E shift the curve in the plane, while F sets the level set value.

one cone, every curve 🖖

Point a flashlight straight at a wall and its beam draws a circle; tilt it and the circle stretches into an ellipse, opens into a parabola, and finally splits into a hyperbola. All four are slices of the same cone of light. This tool does the reverse: from the six coefficients A–F it reads off which slice you have drawn. A circle is simply the special ellipse where A = C and B = 0.

orbits pick a conic by their energy 🖖

Newton showed that any object moving under inverse-square gravity follows a conic with the Sun at one focus, and the sign of its total energy decides which: negative energy binds it into an ellipse, exactly zero gives a parabolic escape, positive energy opens a hyperbola. That is how astronomers knew ʻOumuamua (2017) and comet 2I/Borisov (2019) came from beyond the solar system — their orbits were measurably hyperbolic (e > 1), so nothing could pull them back.

Conic Sections Theory — Curves of the Cone & Plane

Classifying Conic Sections: From Cone Cuts to Equations

Every conic section is formed by intersecting a double-nappe cone with a plane. The plane's tilt defines the geometric family, while its algebraic equation is classified by the discriminant Δ = B² - 4AC and eccentricity e.

Circle — Constant Radius & Zero Eccentricity x² + y² = r²

Ellipse — Two Foci & Bounded Orbit x²/a² + y²/b² = 1

Parabola — Equal Distance to Focus & Directrix y = a(x − h)² + k

Hyperbola — Two Unbounded Branches & Asymptotes x²/a² − y²/b² = 1

Degenerate Conics — Apex Intersection Ax² + Cy² = 0

Rotated Conics — Cross-Term Alignment tan(2θ) = B / (A − C)

Circle — Constant Radius & Zero Eccentricity

Given conditions: Discriminant Δ < 0, A = C ≠ 0, B = 0 (or e = 0)

Equation: x² + y² = r²

Worked example: x² + y² = 9 ⟹ Circle centered at (0,0) with radius r = 3

Open this conic: circle

Circle — Constant Radius & Zero Eccentricity. Circle: A cutting plane perpendicular to the cone's axis of symmetry. Discriminant Δ < 0, A = C ≠ 0, B = 0 (or e = 0) — Circle: A cutting plane perpendicular to the cone's axis of symmetry.

Ellipse — Two Foci & Bounded Orbit

Given conditions: Discriminant Δ < 0, A ≠ C or B ≠ 0 (0 < e < 1)

Equation: x²/a² + y²/b² = 1

Worked example: 4x² + 9y² = 36 ⟹ Semi-major axis a = 3, semi-minor axis b = 2

Open this conic: tilted ellipse

Ellipse — Two Foci & Bounded Orbit. Ellipse: A plane cutting through one nappe at a shallow tilt angle. Discriminant Δ < 0, A ≠ C or B ≠ 0 (0 < e < 1) — Ellipse: A plane cutting through one nappe at a shallow tilt angle.

Parabola — Equal Distance to Focus & Directrix

Given conditions: Discriminant Δ = B² - 4AC = 0 (e = 1)

Equation: y = a(x − h)² + k

Worked example: y - x² = 0 ⟹ Vertex at (0,0), focus at (0, 0.25)

Open this conic: parabola

Parabola — Equal Distance to Focus & Directrix. Parabola: A plane cutting parallel to the generator side of the cone. Discriminant Δ = B² - 4AC = 0 (e = 1) — Parabola: A plane cutting parallel to the generator side of the cone.

Hyperbola — Two Unbounded Branches & Asymptotes

Given conditions: Discriminant Δ = B² - 4AC > 0 (e > 1)

Equation: x²/a² − y²/b² = 1

Worked example: x² - y² = 4 ⟹ Two branches opening horizontally with asymptotes y = ±x

Open this conic: hyperbola

Hyperbola — Two Unbounded Branches & Asymptotes. Hyperbola: A steep plane cutting through both upper and lower nappes. Discriminant Δ = B² - 4AC > 0 (e > 1) — Hyperbola: A steep plane cutting through both upper and lower nappes.

Degenerate Conics — Apex Intersection

Given conditions: Plane passes directly through the central cone vertex

Equation: Ax² + Cy² = 0

Worked example: x² + y² = 0 (Point) or x² - y² = 0 (Two lines y = ±x)

Open this conic: degenerate lines

Degenerate Conics — Apex Intersection. Degenerate Conic: Slicing through the central apex collapses the curve. Plane passes directly through the central cone vertex — Degenerate Conic: Slicing through the central apex collapses the curve.

Rotated Conics — Cross-Term Alignment

Given conditions: Mixed term B ≠ 0 in Ax² + Bxy + Cy² + Dx + Ey + F = 0

Equation: tan(2θ) = B / (A − C)

Worked example: 4x² + xy + 2y² = 16 ⟹ Rotated ellipse with tilted axes

Open this conic: near parabola

Rotated Conics — Cross-Term Alignment. Rotated Conic: Cross-terms tilt the major and minor axes in the xy-plane. Mixed term B ≠ 0 in Ax² + Bxy + Cy² + Dx + Ey + F = 0 — Rotated Conic: Cross-terms tilt the major and minor axes in the xy-plane.

References (1)

Insight block 3 — the hyperbolic orbit that proved an object came from outside: K. J. Meech et al., "A brief visit from a red and extremely elongated interstellar asteroid." Nature 552, 378–381, 2017 — 1I/ʻOumuamua, whose positive orbital energy is what identified it.

Problem solved in full

The angle the crooked ellipse is tilted by, its two axes and eccentricity 5 steps
4x² + xy + 2y² = 16 is an ellipse, and the xy term means it sits crooked. Find the angle it is tilted by, its two axes and its eccentricity — none of which this panel prints.
1. The discriminant settles the family first. It is negative, so the curve is an ellipse, and no amount of rotating or shifting can change that.
  
  $Δ = B^{2} - 4 A C = 1 - 4 (4) (2) = 1 - 32 = - 31 < 0$
2. Write the quadratic part as a symmetric matrix, with the xy coefficient split between the two off-diagonal entries. Its determinant is 7.75, and the discriminant is exactly −4 times it — the same test in different notation.
  
  $M = (A B /2 B /2 C) = (4 0.5 0.5 2), det M = 8 - 0.25 = 7.75 = - \frac{Δ}{4}$
3. Rotating the axes by θ changes B, and the rotation that removes it entirely satisfies tan 2θ = B/(A − C). Solving gives a tilt of about 13.3°.
  
  $tan 2 θ = \frac{B}{A - C} = \frac{1}{2} ⟹ θ = 13.282 5^{\circ}$
4. In the rotated frame the equation has no cross term, and the two new coefficients are the eigenvalues of that matrix. The ellipse is then in its familiar form.
  
  $λ = \frac{6 \pm 5}{2} = 4.118, 1.882 ⟹ 4.118 x^{'2} + 1.882 y^{'2} = 16$
5. The semi-axes follow, and with them the eccentricity: 0.737. That is a decidedly elongated ellipse — the short axis is 68% of the long one, and no planet in the solar system is on an orbit anywhere near that shape.
  
  $a = \frac{16}{1.882} = 2.916, b = \frac{16}{4.118} = 1.971, e = 1 - \frac{b ^{2}}{a ^{2}} = 0.737$
Answer

The tool prints Δ = −31 and classifies it as an Ellipse. Step 2 is why that classification can be trusted. Rotating the axes changes A, B and C individually, but it cannot change the determinant of the matrix they form, and the discriminant is nothing but that determinant times −4. So the ellipse–parabola–hyperbola boundary is a property of the curve rather than of the coordinates you happened to draw it in — which is exactly what a classification has to be to be worth having.

$Δ = - 4 det M, and det M survives rotation$

By the 42Math team · Updated 2026-08-17

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

All Mathematics lessons →

Example problems

circle - x^2 + y^2 = 9 is a circle, a special case of ellipse.
tilted ellipse - Nonzero xy term produces a rotated ellipse.
parabola - y = x^2 gives parabola classification (delta=0).
hyperbola - x^2 - y^2 = 4 is a two-branch hyperbola.
degenerate lines - Δ = 4 here, exactly as for the hyperbola preset — and the classification reads Degenerate. The only difference between the two is F: 0 against −4, and F does not appear in B² − 4AC at all. x² − y² = 0 factors as (x − y)(x + y) = 0, the two asymptotes on their own, which is what a hyperbola becomes once you let it close on them. The discriminant names the family; it cannot tell you whether the curve survived.
near parabola - Δ = 2² − 4(1)(1) = 0 exactly, so the classification reads Parabola, not a near one. Grouped, the equation is (x + y)² = 4y: the xy term has turned the parabola off the axes without changing what it is. Nudge B down by a tenth and Δ goes negative and the curve closes into an ellipse; nudge it up instead and it opens into a hyperbola. The whole family turns on that one number.