A Conic Equation Calculator helps determine the type and characteristics of a conic section based on its equation. Conic sections, including circles, ellipses, parabolas, and hyperbolas, are integral to geometry and have applications in physics, engineering, and astronomy. The calculator simplifies the process of classifying conic sections and extracting critical parameters like the center, radius, axes, vertices, and foci.
Formula of Conic Equation Calculator
General Conic Equation
The general equation for a conic section is:
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
Where:
- A, B, C, D, E, and F are constants that define the type and orientation of the conic.
Step 1: Identify the Conic Type
The type of conic section is determined by the discriminant (Δ):
Discriminant (Δ) = B^2 – 4AC
- If Δ < 0 and A = C, the conic is a circle.
- If Δ < 0 and A ≠ C, the conic is an ellipse.
- If Δ = 0, the conic is a parabola.
- If Δ > 0, the conic is a hyperbola.
Step 2: Standard Forms for Conics
- Circle
(x – h)^2 + (y – k)^2 = r^2- Center: (h, k)
- Radius: r
- Ellipse
((x – h)^2 / a^2) + ((y – k)^2 / b^2) = 1- Center: (h, k)
- Major axis: 2a
- Minor axis: 2b
- Parabola
- Vertical: (x – h)^2 = 4p(y – k)
- Horizontal: (y – k)^2 = 4p(x – h)
- Vertex: (h, k)
- Focus: Distance p from the vertex
- Hyperbola
- ((x – h)^2 / a^2) – ((y – k)^2 / b^2) = 1
- ((y – k)^2 / b^2) – ((x – h)^2 / a^2) = 1
- Center: (h, k)
- Distance between foci: 2c, where c = sqrt(a^2 + b^2)
Step 3: Transform the General Equation
- Rearrange terms to group x and y.
- Complete the square to rewrite the equation in standard form.
- Identify key parameters (h, k, a, b, r, p) from the standard form.
General Table for Common Conics and Characteristics
| Conic Type | General Equation Form | Key Parameters |
|---|---|---|
| Circle | (x – h)^2 + (y – k)^2 = r^2 | Center: (h, k), Radius: r |
| Ellipse | ((x – h)^2 / a^2) + ((y – k)^2 / b^2) = 1 | Center: (h, k), Axes: 2a, 2b |
| Parabola | (x – h)^2 = 4p(y – k) or (y – k)^2 = 4p(x – h) | Vertex: (h, k), Focus: p |
| Hyperbola | ((x – h)^2 / a^2) – ((y – k)^2 / b^2) = 1 | Center: (h, k), Foci: 2c |
Example of Conic Equation Calculator
Problem: Classify the Conic and Extract Parameters
Equation: 4x^2 + 9y^2 – 36 = 0
- Rearrange:
4x^2 + 9y^2 = 36
Divide through by 36:
(x^2 / 9) + (y^2 / 4) = 1 - This is the standard form of an ellipse:
- Center: (0, 0)
- Major axis: 2a = 6 (a = 3)
- Minor axis: 2b = 4 (b = 2)
Most Common FAQs
A conic section is a curve obtained by intersecting a plane with a double-napped cone. The types include circles, ellipses, parabolas, and hyperbolas.
The discriminant (Δ = B^2 – 4AC) classifies conics. If Δ < 0, the conic is a circle or ellipse; if Δ = 0, it is a parabola; if Δ > 0, it is a hyperbola.
Yes, for rotated conics where B ≠ 0, the calculator includes steps to account for rotation and redefines the equation accordingly.