About this calculator
How to identify and analyze conic sections? Conic sections include circles, ellipses, parabolas and hyperbolas, which are curves obtained by truncating a conic surface. The general equation of a conic section is Ax²+Bxy+Cy²+Dx+Ey+F=0. The type of curve can be determined by the discriminant: when B²-4AC<0, it is an ellipse, when it is equal to 0, it is a parabola, and when it is greater than 0, it is a hyperbola.
Conic sections are ubiquitous in nature and engineering. The orbits of planets around the sun are ellipses, parabolas are the trajectories of projectile motion, and hyperbolas appear in hyperbolic navigation systems. In optics, parabolic mirrors focus parallel light, and elliptical mirrors have two focal points. In architecture, arch bridges often adopt a parabolic shape.
Our conic calculator identifies types of conic sections, solves standard equations, and calculates key parameters (such as focus, vertex, eccentricity, etc.). Supports conversion between general equations and standard equations, providing detailed analysis and geometric illustrations.
What it calculates
The conic sections calculator identifies and computes key parameters for circles, ellipses, parabolas, and hyperbolas.
Formula
- Circle: (x - h)^2 + (y - k)^2 = r^2.
- Ellipse: (x - h)^2/a^2 + (y - k)^2/b^2 = 1.
- Parabola: (y - k)^2 = 4p(x - h) or (x - h)^2 = 4p(y - k).
- Hyperbola: (x - h)^2/a^2 - (y - k)^2/b^2 = 1.
Inputs
- The equation or standard-form parameters.
- Known center, focus, vertex, or axis lengths.
Example
| Equation | Type | Key information |
|---|---|---|
| x^2 + y^2 = 9 | Circle | Radius 3 |
| x^2/9 + y^2/4 = 1 | Ellipse | Semi-axes 3 and 2 |
| y^2 = 8x | Parabola | p = 2 |
How to interpret the result
The result helps identify shape, position, and opening direction. Standard form is the easiest way to read center, axes, foci, and vertices.
Common mistakes
- Signs of squared terms determine the conic type.
- General form usually needs completing the square.
- Ellipse and hyperbola denominators need careful axis interpretation.
How to use
Using the conic section calculator is very simple. Just enter the equation or parameters.
**Method 1: Enter the general equation** Enter the coefficient of Ax²+Bxy+Cy²+Dx+Ey+F=0, and the calculator automatically recognizes the curve type and converts it to a standard equation.
**Example 1:** The equation x²+4y²-2x-16y+13=0. The formula yields (x-1)²+4(y-2)²=4, that is, (x-1)²/4+(y-2)²/1=1. This is an ellipse with center (1,2), major axis 2, and minor axis 1.
**Method 2: Enter the parameters of the standard equation** Select the curve type (ellipse, parabola, hyperbola), enter parameters (such as center, focus, vertex, etc.) to get the standard equation.
**Example 2:** Ellipse, center (0,0), major semi-axis a=5, minor semi-axis b=3. Equation: x²/25+y²/9=1. Focus (±4,0), eccentricity e=4/5=0.8.
Main features
• Curve recognition: automatically recognizes conic section types • Standard Equation: Convert to standard equation form • Key parameters: Calculate focus, vertex, eccentricity, directrix, etc. • Geometric graphics: drawing conic sections • Property analysis: analyze the geometric properties of the curve • Equation conversion: general equation ↔ standard equation • Rotation transformation: processing equations containing xy terms • Tangent equation: find the tangent line through a point on the curve • Batch analysis: supports analysis of multiple curves • Totally free: no registration required, use anytime
Use cases
• Analytical geometry learning: students learn about conic sections • Astronomy: Analysis of planetary orbits (elliptical) • Physics: Projectile trajectories (parabolas) • Optical design: parabolic mirror, elliptical mirror • Architectural design: curved design of arch bridges and domes • Navigation system: hyperbolic navigation and positioning • Exam Preparation: Quick Analysis of Conic Sections • Teaching aid: teacher explains conic sections • Engineering design: curve trajectory design • Computer Graphics: Drawing Conic Sections