1. What is a Conic Equation?

A conic equation describes curves formed by the intersection of a plane with a cone. The general form is Ax² + Bxy + Cy² + Dx + Ey + F = 0, where different values of coefficients produce circles, ellipses, parabolas, or hyperbolas.

2. How Does the Calculator Work?

The calculator uses five points to determine the coefficients of the conic equation:

By substituting the coordinates of five points into this equation, we create a system of five equations that can be solved for the six coefficients (with one degree of freedom).

3. Types of Conic Sections

Circle: A = C, B = 0
Ellipse: B² - 4AC < 0
Parabola: B² - 4AC = 0
Hyperbola: B² - 4AC > 0

4. Using the Calculator

Tips: Enter five distinct points (x,y coordinates). The points should not be collinear and should uniquely determine a conic section. For best results, choose points that clearly define the curve.

5. Frequently Asked Questions (FAQ)

Q1: Why five points?
A: A general conic equation has six coefficients, but only five are independent (the equation can be multiplied by any nonzero constant). Five points provide enough information to determine these coefficients up to a scaling factor.

Q2: What if points are collinear?
A: If three or more points are collinear, they may not uniquely define a conic section, and the calculation may fail or produce degenerate results.

Q3: Can I use this for any conic section?
A: Yes, this method works for circles, ellipses, parabolas, and hyperbolas, as long as the points properly define the curve.

Q4: How accurate is the calculation?
A: Accuracy depends on the precision of input points and the numerical stability of the matrix operations used to solve the system of equations.

Q5: What if I get an error?
A: This may indicate that the points don't properly define a conic section, or there may be numerical issues with the calculation. Try different points or check for collinearity.