Equation of a Conic Section Through Five Points
Conic Section Equation Calculator & and Dynamic 5-Point Conic Applet
A conic section is a curve generated by slicing a double naped cone with a plane. The four conic section curves are the circle, ellipse, parabola, and hyperbola. Circles and ellipses are formed when a plane cuts the cone at an angle that is less than the angle of the cone's sides. A parabola is formed by cutting the cone at an angle equal to the cone's sides, that is, the plane is parallel to the side.
A hyperbola occurs when the plane is parallel to the vertical axis of the cone. Hyperbolic curves consist of two disjoint pieces that are mirror images. When the plane cuts straight through the vertical axis, the hyperbola degenerates into two intersecting lines.
The four curves are show in the figure below:
General Equation of a Conic Section
In Cartesian coordinates, the general equation of a conic section isAx2 + Bxy + Cy2 + Dx + Ey + F = 0,
where not all of A, B, and C are equal to zero. Whether the curve is a hyperbola, parabola, ellipse, or circle depends on the value of the discriminant, B2 - 4AC. The three cases are
B2 - 4AC > 0, hyperbola
B2 - 4AC = 0, parabola
B2 - 4AC < 0, ellipse or circle (circle only if B = 0 and A = C)
Given five distinct points in the xy-plane, no more than three of which are collinear, there is a unique conic section that passes through the points. In the applet below, drag the points to see how the shape of a conic section changes according to the distribution of the five points.
How to Find A, B, C, D, E, and F
Suppose the coordinates of the five points are (m, n), (p, q), (r, s), (u, v), and (w, z). The equation of the conic section is found by calculating the determinant of a 6-by-6 matrix and setting it equal to 0. The matrix equation is:
The coefficients A, B, C, D, E, and F are found from the minors of the the larger matrix. For example, suppose you want to find the equation of the conic section through the points (3, 3), (2, -1), (1, -2), (-2, 1), and (-3, -3). Computing the matrix determinant gives you conic equation
-1188x2 + 1404xy - 1188y2 + 0x + 0y + 8748 = 0.
Using a scientific calculator with GCD function, you can reduce the coefficients of the equation and simplify it as
11x2 - 13xy + 11y2 = 81,
which represents an ellipse centered at the origin. You can use the conic section calculator above the find the equation of a conic curve when you input the coordinates of five points.
More Fun with Conic Sections
The applet below contains eight points, with 24 five-point subsets that define 24 different conic sections. Move the points around inside the window to make colorful designs with overlapping circles, ellipses, parabolas, and hyperbolas.© Had2Know 2010
Applet created with GeoGebra.