Conic Sections Calculator – Guide & Formulas
Analyze and calculate properties of conic sections: circles, ellipses, parabolas, and hyperbolas. Get equations, foci, vertices, and more.
The Conic Sections Calculator identifies and analyzes circles, ellipses, parabolas, and hyperbolas from their general or standard-form equations. Enter the coefficients of the general equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 to determine the type and properties.
Key Takeaway
Use the free Conic Sections Calculator to analyze and calculate properties of conic sections: circles, ellipses, parabolas, and hyperbolas. get equations, foci, vertices, and more. Get instant results with step-by-step explanations.
How to Use the Conic Sections Calculator
- Enter the coefficients A, B, C, D, E, and F from the general conic equation.
- The calculator identifies the conic type automatically.
- Review the standard form, key properties (foci, vertices, axes), and eccentricity.
- View the step-by-step conversion from general to standard form.
The Formula
Variable Definitions
- A, C: Coefficients of x² and y² terms; their relationship determines the conic type
- B: Coefficient of xy cross-term; B=0 means axes are aligned with coordinate axes
- D, E: Linear coefficients affecting center/vertex position
- F: Constant term
- e: Eccentricity: e=0 (circle), 0<e<1 (ellipse), e=1 (parabola), e>1 (hyperbola)
Example: Identifying x²/4 + y²/9 = 1
Determine the conic type and find its properties.
- Step 1: Identify the equation form: x²/4 + y²/9 = 1. Both terms are positive with different denominators.
- Step 2: This is an ellipse since both x² and y² terms are positive with different coefficients.
- Step 3: Since 9 > 4, the major axis is along the y-axis. a² = 9, b² = 4.
- Step 4: Vertices: (0, ±3). Co-vertices: (±2, 0).
- Step 5: Foci: c² = a² − b² = 9 − 4 = 5, so c = √5. Foci at (0, ±√5).
Frequently Asked Questions
What are the four types of conic sections?
The four conic sections are: circle (B²−4AC < 0, A=C), ellipse (B²−4AC < 0, A≠C), parabola (B²−4AC = 0), and hyperbola (B²−4AC > 0).
How do I identify a conic from its equation?
Compute the discriminant Δ = B² − 4AC. If Δ < 0 and A=C, it is a circle. If Δ < 0 and A≠C, ellipse. If Δ = 0, parabola. If Δ > 0, hyperbola.
What is eccentricity?
Eccentricity (e) measures how much a conic deviates from a circle. Circle: e=0, Ellipse: 0<e<1, Parabola: e=1, Hyperbola: e>1. It is the ratio of focal distance to semi-major axis.
What are the foci of an ellipse?
The foci are two special points inside the ellipse such that the sum of distances from any point on the ellipse to both foci is constant (equals 2a). They are located at (±c, 0) where c²=a²−b².
How is a circle a special ellipse?
A circle is an ellipse where the two foci coincide at the center (c=0, e=0), making both axes equal (a=b). The standard equation x²+y²=r² is a special case of the ellipse equation.
What is the directrix of a parabola?
The directrix is a fixed line such that any point on the parabola is equidistant from the focus and the directrix. For y²=4ax, the directrix is x=−a.
What are the asymptotes of a hyperbola?
Asymptotes are lines the hyperbola approaches but never touches. For x²/a²−y²/b²=1, the asymptotes are y=±(b/a)x. They form an X through the center.
How do I convert general to standard form?
Complete the square for both x and y terms. Group x terms, complete the square, then do the same for y. Rearrange to get the standard form on one side.
What is the latus rectum?
The latus rectum is a line segment through the focus perpendicular to the major axis. For an ellipse, its length is 2b²/a. For a parabola, it equals 4a.
How do conic sections relate to real life?
Orbits are ellipses (Kepler), parabolic mirrors focus light, hyperbolic shapes appear in cooling towers, and circles are everywhere in wheels and gears.
What is the difference between a hyperbola and a parabola?
A hyperbola has two branches and two foci, approaching asymptotes. A parabola has one branch, one focus, and opens infinitely without approaching any asymptote.
How is a conic section formed?
Conic sections are formed by intersecting a plane with a double cone. The angle of the plane determines the type: horizontal gives a circle, angled gives ellipse, parallel to slant gives parabola, steep gives hyperbola.
What is the semi-latus rectum?
The semi-latus rectum is half the latus rectum length. For an ellipse, it is b²/a. For a hyperbola, it is b²/a. It is used in the polar equation of conics.
Can a conic section be degenerate?
Yes, degenerate conics occur when the plane passes through the apex of the cone: a point (degenerate circle), a line (degenerate parabola), or two intersecting lines (degenerate hyperbola).
How do I find the center of a conic?
For centered conics (circle, ellipse, hyperbola), the center is found by solving ∂F/∂x = 0 and ∂F/∂y = 0 simultaneously. This gives x₀ = (2CD−BE)/(B²−4AC), y₀ = (2AE−BD)/(B²−4AC).
What is the reflective property of parabolas?
Any ray parallel to the axis of a parabola reflects off the surface and passes through the focus. This property is used in satellite dishes, headlights, and solar concentrators.
How does eccentricity affect the shape?
Low eccentricity (near 0) means the shape is nearly circular. As eccentricity approaches 1 (for ellipses), the shape becomes more elongated. For hyperbolas, higher eccentricity means wider opening.
What is the polar equation of a conic?
r = ep/(1 ± e cos θ) or r = ep/(1 ± e sin θ), where e is eccentricity and p is the semi-latus rectum. This unified form represents all conic types.
How are conic sections used in astronomy?
Planetary orbits are ellipses with the Sun at one focus. Comet orbits can be parabolic or hyperbolic. Hyperbolic trajectories are used for spacecraft slingshot maneuvers.
What is the discriminant of a conic?
The discriminant is Δ = B² − 4AC from the general equation. It determines the conic type without completing the square: negative for circle/ellipse, zero for parabola, positive for hyperbola.