Conic Equation Calculator

Enter the values to use basic and advanced conic equation calculator today!

Conic Equation Calculator

Coefficient of x² (A) Coefficient of xy (B) Coefficient of y² (C) Coefficient of x (D) Coefficient of y (E) Constant term (F) Conic Type

Calculate Reset

The Conic Equation Calculator is a tool used to analyze the properties of conic sections, which are geometric curves formed by intersecting a plane with a cone. This calculator helps determine the type of conic section based on the coefficients of the general conic equation.

Formula:

The formula is:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

Variables:

Variable	Meaning
A	Coefficient of $x^2$
B	Coefficient of $xy$ (the cross term)
C	Coefficient of $y^2$
D	Coefficient of $x$
E	Coefficient of $y$
F	Constant term

How to Calculate ?

1. Identify the Coefficients:
   First, identify the coefficients $A$, $B$, $C$, $D$, $E$, and the constant $F$ from the given conic equation.

2. Classify the Conic Section:
   Based on the values of $A$, $B$, and $C$, classify the conic section. Here are the general rules:

   • Ellipse: If $A$ and $C$ have the same sign and $B = 0$.
   • Parabola: If either $A$ or $C$ is zero.
   • Hyperbola: If $A$ and $C$ have opposite signs.

3. Find Important Parameters:
   Depending on the type of conic section, you can further derive parameters such as the center, axis of symmetry, vertices, and foci.

4. Solving the Equation:
   To solve the equation, substitute the known values of $x$ and $y$ (if given) into the equation to check for valid points on the conic or graph the conic.

Solved Calculation :

Given Equation:

$$2x^2 + 3xy + 2y^2 – 4x – 5y + 1 = 0$$

Calculation	Instructions
Step 1: Identify coefficients: A = 2, B = 3, C = 2, D = -4, E = -5, F = 1	Identify coefficients from the equation.
Step 2: Classify the conic: Since $A$ and $C$ are positive and $B \neq 0$, this represents a rotated ellipse.	Classify based on $A$ $B$, and $C$

Answer:
The given equation represents a rotated ellipse.

What is Conic Equation :

A Conic Equation Calculator is a powerful  for solving and understanding the equations of conic sections, including circles, ellipses, parabolas, and hyperbolas. Conic sections are derived by slicing a cone at different angles, resulting in various shapes with distinct equations.  This calculator helps users input their equation and instantly convert it to the conic form or find key properties like the center, radius, and vertices.

Using a Conic Equation Calculator allows for a deep understanding of how these equations describe geometric shapes and their transformations, such as rotations. This calculator makes it easy the the classification of  a conic section by analyzing its general equation and determining whether it represents a circle, ellipse, parabola, or hyperbola.

Whether you’re working on graphing conic sections, solving for specific parameters, or converting between general and standard forms, this calculator provides quick solutions with detailed steps, perfect for students and professionals in mathematics and engineering.

Final Words:

Conic sections, such as circles, ellipses, parabolas, and hyperbolas, have distinct equations that describe their shape and properties. It is as essential for students, teachers as for  professionals who need to work with conic sections in geometry or calculus.