Conic Sections

Conic sections are a group of geometric shapes from nonlinear equations that can be graphed in a two-dimensional plane. These shapes do not contain any straight lines. They are curvilinear which means consisting of curved lines. The conic section equations all use the power function x squared which is the reason for the rounded shapes when graphed. The four basic conic sections that can be formed in a three-dimensional geometric space are the circle, the ellipse, the parabola and the hyperbola. The shapes are called conic sections because they can be formed by taking two inverted cones connected at their peaks and intersecting them with a two-dimensional plane. The circle is an enclosed two-dimensional shape with an even radius. The radius is the distance from the center to the edge of the circle. The ellipse is an elongated circle created with two focal points where the combined distance from them to the points on the ellipse are a constant sum. The parabola has a vertex point at its center, a focus point and two lines that extend off curving into positive infinity or negative infinity. These lines extend along directional lines called asymptotes. Quadratic functions form parabolas when mapped on the Cartesian Plane. The hyperbola is similar to the parabola, but has pair of two lines that all four curve along asymptotes off to infinity with two two focal points and two vertices.

A circle:
If x² + y² = b²,
then the radius is equal to b
and the center is at the point (0, 0).

An ellipse:
If x²/a² + y²/b² = 1,
then the center is at the point (0, 0)
and the intercept point for y is at (b, 0).

A parabola:
If y = ax² and a is positive,
then the vertex is (0, 0)
with two lines extending upward to positive infinity.

A hyperbola:
If x²/a² - y²/b² = 1,
then the vertices are (a, 0) and (-a, 0),
with four lines extending upward to infinity
and downward to negative infinity
along the asymptotes ax/b and -ax/b.

Math Mysteries: Each conic section can be used in the field of optics to reflect light differently. The circle reflects it to the center, the ellipse reflects from one focal point to the other, a parabola as a parallel beam and a hyperbola goes from one focal point through the other.