Conic sections and their equations

Connect conic equations to geometric definitions

Problem types

Equations of circles
Completing the square
Distance and midpoints in the complex plane · Useful idea
The points z whose distance from w is a fixed number r form a circle, the first conic section.
Where a line meets a parabola · Useful idea
Reading where a line crosses a circle or ellipse counts and locates the solutions of a system with a conic.
Solving a linear and a nonlinear equation together · Useful idea
Finding where a line meets a circle or other conic solves a nonlinear system.
A parabola as points equally far from a focus and a line · Earlier idea
A parabola is the points equally far from a focus and a directrix. Ellipses and hyperbolas extend the idea: constant sum or difference of distances to two foci.
A tangent from a point outside a circle · Earlier idea
A tangent to a conic touches it at one point, extending the tangent to a circle to ellipses, parabolas and hyperbolas.
Cross-sections of solids · Useful idea
Slicing a cone with a plane gives a circle, an ellipse, a parabola or a hyperbola, the conic sections.

None listed.

Motion described by a parameter · Related idea
A circle or ellipse can be traced by x = h + a cos t, y = k + b sin t, which satisfies the conic's standard equation because cos² t + sin² t = 1.
Polar and Cartesian coordinates · Related idea
Conics have polar equations with a focus at the origin, so the same curve has Cartesian and polar forms.
Asymptotes, holes, and intercepts · Related idea
The function y = 1/x is a hyperbola with the axes as asymptotes, a conic section.
Vectors and their components · Related idea
A conic is defined by distances from foci, which are lengths of vectors from the foci to the point.
Cross-sections and solids of revolution · Related idea
Rotating a parabola about its axis gives a paraboloid, and slicing a cone gives the conic curves.

Open the interactive map to explore lessons and skill connections.