Conic Sections Calculator

Analyze circles, parabolas, ellipses, hyperbolas, focus-directrix definitions, eccentricity, tangent lines, and general quadratic equations with guided steps.

Inputs

Uses standard circle form with exact center, radius, area, circumference, and graph bounds.

Completes the square to find center and radius from general circle form.

Finds focus, directrix, axis, latus rectum, and vertex-form equation.

Completes the square to convert a quadratic parabola into vertex form.

Finds vertices, co-vertices, foci, eccentricity, area, and standard equation.

Finds foci, vertices, asymptotes, eccentricity, and standard equation.

Classifies Ax² + Bxy + Cy² + Dx + Ey + F = 0 using the discriminant.

Builds a conic from the equation PF = e · distance to directrix.

Computes focal distance, eccentricity, and latus rectum length.

Finds the tangent line to a circle at a point on the circle.

Step view
Worked Examples
Circlecenter (2,−1), r = 5General circlex²+y²−6x+4y−12=0Parabolaopens up from (1,−2), p = 3Parabolay = ¼x² − x + 2Ellipsecenter (0,0), a = 6, b = 4Hyperbolacenter (0,0), a = 4, b = 3Classifier9x² + 4y² −18x +16y −11=0Focus-directrixF(2,0), x = −2, e = 1Eccentricityellipse with a = 5, b = 3Tangentcircle r = 5 at (3,4)

Tools
Circlecenter, radius, area, circumference
General circlecomplete the square
Parabolafocus, directrix, latus rectum
Ellipsefoci, vertices, eccentricity
Hyperbolaasymptotes and branches
Classifieruse discriminant B²−4AC
Focus-directrixbuild equation from definition
Tangentradius-normal tangent line
Input Tips
  • Fractions like 3/4 are accepted.
  • Use positive a, b, and r values.
  • The classifier accepts rotated conics through Bxy.
  • For tangents, the chosen point must lie on the circle.
  • When e = 1, focus-directrix form gives a parabola.
  • Ellipse eccentricity is less than one.
  • Hyperbola eccentricity is greater than one.
CalcPro Conic Sections Calculator © 2026 · PHP Step Engine + Canvas Conic Graphs + KaTeX

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