Conic Section Calculator

Circle

A circle is a set of all points in a plane that are equidistant from a fixed point called the center. It's a perfectly round shape, often seen in wheels, coins, and orbits.

(x - h)² + (y - k)² = r²

• Center: (h, k): This is the fixed point from which all points on the circle are equally distant. It defines the circle's position on a coordinate plane.
• Radius: r: The constant distance from the center to any point on the circle. It determines the size of the circle.
• Area: πr²: The amount of space enclosed by the circle.
• Circumference: 2πr: The distance around the circle.

Ellipse

An ellipse is a set of all points in a plane such that the sum of the distances from two fixed points (called foci) is constant. It looks like a stretched or flattened circle, commonly seen in planetary orbits and architectural designs.

((x - h)²/a²) + ((y - k)²/b²) = 1

• Center: (h, k): The midpoint of both the major and minor axes. It defines the ellipse's position.
• Semi-major axis: a: Half the length of the longest diameter of the ellipse. It determines the ellipse's overall size along its longer dimension.
• Semi-minor axis: b: Half the length of the shortest diameter of the ellipse. It determines the ellipse's overall size along its shorter dimension.
• Area: πab: The amount of space enclosed by the ellipse.
• Eccentricity: e = √(1 - b²/a²): A measure of how "stretched" or "flattened" an ellipse is. An eccentricity of 0 means it's a perfect circle, while values closer to 1 indicate a more elongated shape.

Parabola

A parabola is a set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). Parabolas are often seen in the paths of projectiles, satellite dishes, and car headlights.

Vertical: (x - h)² = 4p(y - k)

Horizontal: (y - k)² = 4p(x - h)

• Vertex: (h, k): The turning point of the parabola, where it changes direction.
• Focus: p units from vertex: A fixed point that defines the shape of the parabola. All points on the parabola are equidistant from this focus and the directrix.
• Directrix: p units from vertex: A fixed line that defines the shape of the parabola. All points on the parabola are equidistant from this line and the focus.
• Eccentricity: e = 1: For a parabola, the eccentricity is always 1, indicating its unique open, U-shaped curve.

Hyperbola

A hyperbola is a set of all points in a plane such that the absolute difference of the distances from two fixed points (foci) is constant. It consists of two separate, open curves, often found in navigation systems and cooling tower designs.

((x - h)²/a²) - ((y - k)²/b²) = 1

• Center: (h, k): The midpoint of the segment connecting the two foci. It defines the hyperbola's position.
• Transverse axis: 2a: The segment connecting the two vertices of the hyperbola. Its length is 2a.
• Conjugate axis: 2b: The segment perpendicular to the transverse axis, passing through the center. Its length is 2b.
• Eccentricity: e = √(1 + b²/a²): A measure of how "open" the branches of the hyperbola are. For hyperbolas, eccentricity is always greater than 1.
• Asymptotes: y = ±(b/a)x: Lines that the hyperbola approaches but never touches as it extends infinitely. They guide the shape of the hyperbola's branches.

Geometric Properties

• Focus-Directrix Relationship: This fundamental property defines parabolas, ellipses, and hyperbolas as the locus of points where the ratio of the distance to a focus and the distance to a directrix is constant (the eccentricity).
• Reflection Properties:
  • Parabola: Any ray parallel to the axis of a parabola reflects off the curve and passes through its focus. This is why parabolic mirrors are used in telescopes and satellite dishes to collect parallel rays of light or signals at a single point.
  • Ellipse: Any ray originating from one focus of an ellipse reflects off the curve and passes through the other focus. This property is used in "whispering galleries" where sound from one focus can be heard clearly at the other.
  • Hyperbola: A ray directed towards one focus of a hyperbola reflects off the curve as if it came from the other focus. This is applied in certain optical systems and navigation.
• Symmetry: All conic sections exhibit symmetry. Circles and ellipses have two axes of symmetry, parabolas have one, and hyperbolas have two. This symmetry simplifies their analysis and design.
• Tangent Lines: The properties of tangent lines to conic sections are crucial in calculus and physics, describing the instantaneous direction of the curve at any point.

Real-world Applications

• Planetary Orbits (Ellipses): Planets, comets, and asteroids orbit the sun in elliptical paths, with the sun located at one of the foci of the ellipse. This was famously discovered by Johannes Kepler.
• Satellite Dishes (Parabolas): The parabolic shape of satellite dishes allows them to collect weak signals from distant satellites and focus them onto a single receiver, maximizing signal strength.
• Whispering Galleries (Ellipses): Architectural spaces like the National Statuary Hall in the U.S. Capitol are elliptical, allowing a whisper from one focal point to be heard clearly at the other, even across a large room.
• Cooling Towers (Hyperbolas): Many large industrial cooling towers are designed with a hyperbolic cross-section. This shape provides structural strength, allows for efficient cooling, and minimizes material usage.
• Telescopes (Parabolas and Hyperbolas): Reflecting telescopes use parabolic mirrors to gather light and often hyperbolic mirrors to redirect it to an eyepiece or sensor, enabling us to see distant celestial objects.
• GPS Navigation (Hyperbolas): Hyperbolic positioning systems (like LORAN, a precursor to GPS) use the time difference of arrival of signals from multiple transmitters to determine a receiver's location, which lies on a hyperbola.