Conic Sections Calculator
Calculate properties of conic sections (circle, ellipse, parabola, hyperbola)
Category: Mathematics
Conic Sections Calculator Inputs
Conic Sections Calculator Formula
Equation
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
Excel Formula
=POWER(Ax,2)+Bxy+POWER(Cy,2)+Dx+Ey+F=0
Variables
- A — Enter the A value used by the Conic Sections Calculator.
- B — Enter the B value used by the Conic Sections Calculator.
- C — Enter the C value used by the Conic Sections Calculator.
- D — Enter the D value used by the Conic Sections Calculator.
- E — Enter the E value used by the Conic Sections Calculator.
- F — Enter the F value used by the Conic Sections Calculator.
How the Conic Sections Calculator Works
Conic sections are the curves obtained by intersecting a right circular cone with a plane at different angles. These fundamental curves in geometry include circles, ellipses, parabolas, and hyperbolas, each with unique geometric properties and mathematical characteristics. They play crucial roles in astronomy, physics, engineering, and many other fields.
The core relationship is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. Typical inputs include A, B, C, D.
Enter your values in the conic sections calculator above, review the step-by-step solution, and compare against the worked examples below so you can see how each input changes the result. This free online mathematics tool is built for homework, design checks, and professional verification.
Conic Sections Calculator Theory & Explanation
General Form and Classification
The general equation of a conic section is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. The discriminant Δ = B^2 - 4AC determines the type: Δ < 0 (ellipse/circle), Δ = 0 (parabola), Δ > 0 (hyperbola). The rotation angle θ is given by \tan(2θ) = B/(A-C) when A ≠ C.
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \quad \textwhere Δ = B^2 - 4AC
Circle Properties and Equations
A circle is the set of all points equidistant from a center point. Standard form: (x-h)^2 + (y-k)^2 = r^2 where (h,k) is the center and r is the radius. General form: x^2 + y^2 + Dx + Ey + F = 0 with center (-D/2, -E/2) and radius √((D/2)^2 + (E/2)^2 - F). The circle occurs when A = C and B = 0 in the general equation.
\textStandard: (x-h)^2 + (y-k)^2 = r^2 \quad \textGeneral: x^2 + y^2 + Dx + Ey + F = 0
Ellipse Properties and Equations
An ellipse is the set of all points where the sum of distances to two fixed points (foci) is constant. Standard form: (x-h)^2/a^2 + (y-k)^2/b^2 = 1 where (h,k) is the center, a and b are semi-major and semi-minor axes. Foci distance: c^2 = a^2 - b^2 (assuming a ≥ b). Eccentricity: e = c/a < 1. The ellipse occurs when Δ < 0 and A ≠ C or B ≠ 0.
((x-h)^2)/(a^2) + ((y-k)^2)/(b^2) = 1 \quad \textwhere c^2 = a^2 - b^2, \quad e = (c)/(a)
Parabola Properties and Equations
A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). Standard forms: y = a(x-h)^2 + k (vertical) or x = a(y-k)^2 + h (horizontal). Focus: (h, k + 1/(4a)) for vertical parabola. Directrix: y = k - 1/(4a). The parabola occurs when Δ = 0 and represents the boundary between elliptical and hyperbolic behavior.
y = a(x-h)^2 + k \quad \textFocus: (h, k + (1)/(4a)) \quad \textDirectrix: y = k - (1)/(4a)
Hyperbola Properties and Equations
A hyperbola is the set of all points where the absolute difference of distances to two fixed points (foci) is constant. Standard form: (x-h)^2/a^2 - (y-k)^2/b^2 = 1 (horizontal) or (y-k)^2/a^2 - (x-h)^2/b^2 = 1 (vertical). Foci distance: c^2 = a^2 + b^2. Eccentricity: e = c/a > 1. Asymptotes: y = ±(b/a)(x-h) + k for horizontal hyperbola. The hyperbola occurs when Δ > 0.
((x-h)^2)/(a^2) - ((y-k)^2)/(b^2) = 1 \quad \textwhere c^2 = a^2 + b^2, \quad e = (c)/(a)
Discriminant Analysis and Classification
The discriminant Δ = B^2 - 4AC provides a complete classification: Δ < 0 gives ellipses (circles when A = C and B = 0), Δ = 0 gives parabolas, and Δ > 0 gives hyperbolas. When Δ = 0 and the conic is degenerate, it may represent two parallel lines, a single line, or a point. The invariant I_1 = A + C and I_2 = Δ are preserved under rotation.
Δ = B^2 - 4AC \quad \begincases Δ < 0: \textEllipse/Circle \\ Δ = 0: \textParabola \\ Δ > 0: \textHyperbola \endcases
Rotation and Transformation
To eliminate the xy term, rotate the coordinate system by angle θ where \tan(2θ) = B/(A-C). The rotation matrix transforms (x,y) to (x',y') where x = x'\cosθ - y'\sinθ and y = x'\sinθ + y'\cosθ. After rotation, the equation becomes A'x'^2 + C'y'^2 + D'x' + E'y' + F' = 0 with A' = A\cos^2θ + B\sinθ\cosθ + C\sin^2θ.
\tan(2θ) = (B)/(A-C) \quad \textRotation: \beginpmatrix x \\ y \endpmatrix = \beginpmatrix \cosθ & -\sinθ \\ \sinθ & \cosθ \endpmatrix \beginpmatrix x' \\ y' \endpmatrix
Geometric Properties and Applications
Circles have constant curvature and appear in wheels, orbits, and wave fronts. Ellipses model planetary orbits (Kepler's first law) and have applications in optics and acoustics. Parabolas are used in satellite dishes, headlights, and projectile motion. Hyperbolas appear in navigation systems (LORAN), cooling towers, and hyperbolic geometry. Each conic has unique optical properties: circles focus parallel rays to a point, ellipses have two foci, parabolas focus parallel rays to a single focus, and hyperbolas have asymptotic behavior.
\textKepler's 1st Law: (r^2)/(a^2) + (z^2)/(b^2) = 1 \quad \text(elliptical orbits)
Eccentricity and Focus-Directrix Definition
All conic sections can be defined using eccentricity e: e = 0 (circle), 0 < e < 1 (ellipse), e = 1 (parabola), e > 1 (hyperbola). The focus-directrix definition: PF = e · PD where P is any point on the conic, F is the focus, D is the directrix, and e is the eccentricity. This unified definition shows that conic sections are the locus of points with constant ratio of distances to focus and directrix.
PF = e · PD \quad \textwhere e = \begincases 0: \textCircle \\ (0,1): \textEllipse \\ 1: \textParabola \\ >1: \textHyperbola \endcases
Conic Sections Calculator Worked Examples
Worked Example
Inputs
- A: 1
- B: 1
- C: 1
- D: 1
- E: 1
- F: 1
Result: Calculated result shown after submitting the example inputs.
Explanation
This example demonstrates how to enter typical values in the Conic Sections Calculator. The calculator applies the formula to the provided inputs and returns the computed result with any available supporting details.
Circle Example
Inputs
- A: 1
- B: 0
- C: 1
- D: 0
- E: 0
- F: -4
Result: Circle
Explanation
A = 1, C = 1, B = 0, so Δ = 0² - 4(1)(1) = -4 < 0. Since A = C and B = 0, this is a circle: x² + y² = 4 with center (0,0) and radius 2.
Common Conic Sections Calculator Use Cases
- Homework and exam practice
- Engineering and science coursework
- Quick verification of hand calculations
- Calculate properties of conic sections (circle
- Ellipse
Conic Sections Calculator FAQs
What are the real-world applications of conic sections?
Conic sections have extensive applications: Circles in wheels, gears, and circular motion; Ellipses in planetary orbits (Kepler's laws), satellite dishes, and optical systems; Parabolas in satellite dishes, headlights, projectile motion, and suspension bridges; Hyperbolas in navigation systems (LORAN), cooling towers, and hyperbolic geometry. They also appear in architecture, engineering, and physics.
How do I convert from general form to standard form?
To convert Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 to standard form: (1) If B ≠ 0, rotate coordinates using \tan(2θ) = B/(A-C) to eliminate the xy term. (2) Complete the square for x and y terms. (3) Factor and rearrange to get standard form. This reveals the center, vertices, foci, and other geometric properties.
What is the difference between an ellipse and a hyperbola?
An ellipse is a closed curve where the sum of distances to two foci is constant, with eccentricity 0 < e < 1. A hyperbola has two separate branches where the absolute difference of distances to two foci is constant, with eccentricity e > 1. The discriminant distinguishes them: Δ < 0 for ellipse, Δ > 0 for hyperbola. Ellipses are bounded, hyperbolas are unbounded with asymptotes.
When is a conic section degenerate?
A conic section is degenerate when it reduces to simpler geometric objects: (1) A point when A = C and B = 0 with D^2 + E^2 - 4F = 0. (2) A single line when the equation factors as (ax + by + c)^2 = 0. (3) Two parallel lines when it factors as (ax + by + c)(ax + by + d) = 0. (4) Two intersecting lines when it factors as (ax + by + c)(dx + ey + f) = 0.
How do I find the foci of an ellipse or hyperbola?
For ellipse (x-h)^2/a^2 + (y-k)^2/b^2 = 1: foci are at (h ± c, k) where c^2 = a^2 - b^2 (assuming a ≥ b). For hyperbola (x-h)^2/a^2 - (y-k)^2/b^2 = 1: foci are at (h ± c, k) where c^2 = a^2 + b^2. The distance from center to each focus is c.
What is eccentricity and how does it relate to conic sections?
Eccentricity e measures how much a conic section deviates from being circular: e = 0 (circle), 0 < e < 1 (ellipse), e = 1 (parabola), e > 1 (hyperbola). It's defined as e = c/a where c is the distance from center to focus and a is the semi-major axis. Higher eccentricity means more "stretched" or "open" the curve becomes.
How do I find the asymptotes of a hyperbola?
For hyperbola (x-h)^2/a^2 - (y-k)^2/b^2 = 1: asymptotes are y = ±(b/a)(x-h) + k. For (y-k)^2/a^2 - (x-h)^2/b^2 = 1: asymptotes are y = ±(a/b)(x-h) + k. The asymptotes pass through the center (h,k) and have slopes ± b/a or ± a/b depending on orientation.
What is the focus-directrix definition of conic sections?
All conic sections can be defined as the locus of points P where the ratio of distance to focus F and distance to directrix D is constant: PF = e · PD. The constant e is the eccentricity: e = 0 (circle), 0 < e < 1 (ellipse), e = 1 (parabola), e > 1 (hyperbola). This unified definition shows conic sections as curves with constant focus-directrix ratio.
How do I determine the orientation of a conic section?
Orientation depends on the coefficients: (1) For ellipses: if a > b, major axis is horizontal; if b > a, major axis is vertical. (2) For parabolas: if x^2 term exists, opens up/down; if y^2 term exists, opens left/right. (3) For hyperbolas: if x^2/a^2 - y^2/b^2 = 1, transverse axis is horizontal; if y^2/a^2 - x^2/b^2 = 1, transverse axis is vertical. (4) For rotated conics, use the rotation angle θ = (1)/(2)\arctan(B/(A-C)).
What are the optical properties of conic sections?
Conic sections have unique optical properties: (1) Circles: parallel rays reflect to center. (2) Ellipses: rays from one focus reflect to the other focus. (3) Parabolas: parallel rays reflect to the focus (used in satellite dishes and headlights). (4) Hyperbolas: rays directed toward one focus reflect as if coming from the other focus. These properties make conic sections essential in optics and antenna design.
How do I find the vertex of a parabola?
For parabola y = ax^2 + bx + c: vertex is at (-b/(2a), c - b^2/(4a)). For x = ay^2 + by + c: vertex is at (c - b^2/(4a), -b/(2a)). For standard form y = a(x-h)^2 + k: vertex is at (h, k). The vertex is the point where the parabola changes direction and is equidistant from the focus and directrix.
What is the relationship between conic sections and quadratic equations?
Conic sections are the graphs of quadratic equations in two variables. The general form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 represents all conic sections. The discriminant Δ = B^2 - 4AC determines the type: Δ < 0 (ellipse/circle), Δ = 0 (parabola), Δ > 0 (hyperbola). This connects algebraic properties with geometric shapes.
How do I find the center of a conic section?
For conic sections in general form Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0: (1) If B = 0, center is at (-D/(2A), -E/(2C)). (2) If B ≠ 0, first rotate to eliminate xy term, then find center. (3) For standard forms: circles have center (h,k), ellipses have center (h,k), hyperbolas have center (h,k), parabolas have vertex (h,k) (not center). The center is the point of symmetry for the conic section.