Examples for

Conic Sections

Conic sections are curves formed by intersecting a cone and a plane. These curves include circles, ellipses, parabolas and hyperbolas. Wolfram|Alpha can identify a conic section by its equation and can also compute the equation or other properties for a given conic section of a specified type.

Conics

Find information about a conic section through its Cartesian equation.

Plot a conic section and identify its type:

x^2 - 2 y^2 = 1
2x^2 - 3xy + 4y^2 + 6x - 3y - 4 = 0

Ellipses

Find an ellipse and compute its properties.

Compute properties of an ellipse:

ellipse with semiaxes 2,5 centered at (3,0)
ellipse with eccentricity 1/2

Specify an ellipse by its equation:

ellipse with equation (x-2)^2/25 + (y+1)^2/10 = 1

Compute the eccentricity of an ellipse:

eccentricity of an ellipse with semiaxes 12,1

Compute the focal parameter of an ellipse:

focal parameter of an ellipse with semiaxes 4,3

Circles

Find a circle and compute its properties.

Compute properties of a circle:

circle of radius 2 with center (0,-2)
circle with area 30cm^2

Find a circle passing through three points:

circle through (0,0), (1,0), (0,1)

Compute the circumference of a circle:

circumference of a circle with radius 3 miles

Parabolas

Find a parabola and compute its properties.

Compute properties of a parabola:

parabola with focus (3,4) and vertex (-4,5)
parabola (y-2)^2=4x

Compute the directrix of a parabola:

directrix of parabola x^2+3y=16

RELATED EXAMPLES

  • Plane Geometry

    • Hyperbolas

    Find a hyperbola and compute its properties.

    Compute properties of a hyperbola:

    hyperbola with center (100, 200) and focus (110, 180)
    hyperbola semimajor axis 10, focal parameter 2

    Locate the foci of a hyperbola:

    foci of hyperbola with semiaxes 3,4