Conic Equation Calculator

Enter the coefficients of the general second-degree equation to determine the type of conic section it represents. This calculator helps identify whether the equation corresponds to a circle, ellipse, parabola, or hyperbola.

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Conic Section Formula

The general second-degree equation for a conic section is:

Ax² + Bxy + Cy² + Dx + Ey + F = 0

The type of conic is determined by the discriminant (B² – 4AC):

• If B² – 4AC > 0, the conic is a hyperbola.
• If B² – 4AC = 0, the conic is a parabola.
• If B² – 4AC < 0, the conic is an ellipse. If additionally A = C and B = 0, it is a circle.

How to Use the Conic Equation Calculator?

Follow these steps to determine the type of conic section:

1. Input the coefficients A, B, C, D, E, and the constant term F of the general second-degree equation.
2. Click the “Calculate” button to determine the type of conic section.
3. The calculator will display “Circle”, “Ellipse”, “Parabola”, or “Hyperbola” based on the discriminant.
4. Use the “Reset” button to clear all fields and perform a new calculation.

Example Problem:

For the equation 2x² + 4xy + 3y² + x – 5y – 2 = 0, determine the type of conic section.

Coefficients are as follows:

A = 2, B = 4, C = 3, D = 1, E = -5, F = -2

Using the discriminant B² – 4AC, we find that 4² – 4(2)(3) = 16 – 24 = -8, which is less than 0. Therefore, the conic section is an ellipse.