Conic Sections Calculator
Identify a circle, ellipse, parabola or hyperbola from its equation and find the centre, vertices, foci, eccentricity, directrices, latus rectum and asymptotes, with exact surds.
About the Conic Sections Calculator
For Ax² + Cy² + Dx + Ey + F = 0: if A = C it's a circle; if A and C have the same sign, an ellipse; opposite signs, a hyperbola; and if one of them is 0, a parabola. Completing the square turns it into the standard form, from which every feature can be read.
This handles conics whose axes are parallel to x and y (no xy term).
The formula
Ellipse: (x − h)²/a² + (y − k)²/b² = 1, c² = a² − b², e = c/a
- Hyperbola
- (x − h)²/a² − (y − k)²/b² = 1, c² = a² + b²
- Parabola
- (y − k)² = 4p(x − h), focus (h + p, k), directrix x = h − p
- Circle
- (x − h)² + (y − k)² = r²
Worked example
9x² + 4y² − 18x + 16y − 11 = 0
- 9(x − 1)² + 4(y + 2)² = 36
- (x − 1)²/4 + (y + 2)²/9 = 1: an ellipse
- a = 3 (along x = 1), b = 2, c = √5, e = √5/3
Questions people ask
What's the eccentricity?
How stretched the conic is: 0 for a circle, between 0 and 1 for an ellipse, exactly 1 for a parabola, more than 1 for a hyperbola.
How can I tell which conic an equation is?
With no xy term, compare the coefficients A of x² and C of y²: AC > 0 gives an ellipse (a circle when A = C), AC < 0 a hyperbola, and AC = 0 a parabola.