Conic Sections Explorer

Discover the beautiful curves formed by intersecting a plane with a cone

The Magic of Conic Sections

Conic sections are the curves obtained by intersecting a right circular cone with a plane. Depending on the angle of intersection, we get different shapes: circles, ellipses, parabolas, and hyperbolas.

What are Conic Sections?

Conic sections are the curves formed when a plane intersects a double-napped right circular cone. The angle at which the plane intersects the cone determines the type of conic section formed:

  • Circle: Plane perpendicular to the axis of the cone
  • Ellipse: Plane at an angle less than the cone's side (but not perpendicular)
  • Parabola: Plane parallel to the side of the cone
  • Hyperbola: Plane intersects both nappes of the cone

Types of Conic Sections

Circle

(x-h)² + (y-k)² = r²

A circle is formed when the intersecting plane is perpendicular to the axis of the cone.

Example

Equation: (x-2)² + (y+1)² = 9

Center at (2, -1) with radius 3

Ellipse

(x-h)²/a² + (y-k)²/b² = 1

An ellipse is formed when the plane intersects one nappe at an angle greater than the cone's side angle.

Example

Equation: x²/16 + y²/9 = 1

Major axis length 8, minor axis length 6

Parabola

y = a(x-h)² + k

A parabola is formed when the plane is parallel to the side of the cone.

Example

Equation: y = 2x² - 4x + 1

Vertex at (1, -1), opens upwards

Hyperbola

(x-h)²/a² - (y-k)²/b² = 1

A hyperbola is formed when the plane intersects both nappes of the cone.

Example

Equation: x²/9 - y²/4 = 1

Vertices at (±3, 0), asymptotes y = ±(2/3)x

Standard Equations

Standard Equation of a Circle

(x - h)² + (y - k)² = r²

Where:

  • (h, k) is the center of the circle
  • r is the radius of the circle

Example

Find the equation of a circle with center at (2, -3) and radius 5.

Solution: (x - 2)² + (y + 3)² = 25

Standard Equation of an Ellipse

(x - h)²/a² + (y - k)²/b² = 1

Where:

  • (h, k) is the center of the ellipse
  • a is the semi-major axis length
  • b is the semi-minor axis length

Example

Find the equation of an ellipse centered at (-1, 2) with horizontal major axis of length 10 and minor axis of length 6.

Solution: (x + 1)²/25 + (y - 2)²/9 = 1

Standard Equation of a Parabola

y = a(x - h)² + k (vertical)
or
x = a(y - k)² + h (horizontal)

Where:

  • (h, k) is the vertex of the parabola
  • a determines the "steepness" and direction

Example

Find the equation of a parabola with vertex at (1, -2) that opens downward and passes through (3, -6).

Solution: y = -1(x - 1)² - 2

Standard Equation of a Hyperbola

(x - h)²/a² - (y - k)²/b² = 1 (horizontal)
or
(y - k)²/a² - (x - h)²/b² = 1 (vertical)

Where:

  • (h, k) is the center of the hyperbola
  • a is the distance to vertices
  • b determines the asymptotes

Example

Find the equation of a hyperbola centered at (0, 0) with vertices at (±3, 0) and asymptotes y = ±(4/3)x.

Solution: x²/9 - y²/16 = 1

Degenerate Cases

Special Cases of Conic Sections

When the plane passes through the vertex of the cone, we get degenerate conic sections:

Single Point

When the plane intersects only at the vertex

x² + y² = 0

Single Line

When the plane is tangent to the cone

y = mx + b

Intersecting Lines

When the plane intersects both nappes through the vertex

x²/a² - y²/b² = 0

Interactive Practice

Identify the Conic Section