Conic Sections Calculator

About the Conic Sections Calculator

Type any second-degree equation in x and y — the general form Ax² + Bxy + Cy² + Dx + Ey + F = 0, a standard form, or an equation solved for y — and the calculator tells you which conic it is: a circle, an ellipse, a parabola or a hyperbola, or a degenerate case (a single point, two lines, or no real points at all). It completes the square to reach the standard form and lists the centre, vertices, co-vertices, foci, directrices, eccentricity, axes, latus rectum, asymptotes and intercepts, with a graph and every step of the working.

Rotated conics — equations with an xy-term such as xy = 1 or 7x² − 6√3xy + 13y² = 16 — are turned back to their own axes first, and the results are given in the original x and y. Values stay exact (√5/3, 2 + √5, 3√2/2) wherever they can, with decimals beside them. You can also go the other way: build the equation of a circle, parabola, ellipse or hyperbola from what you know, such as its foci and the length of an axis, a focus and a directrix, or three points.

Common uses

• Checking homework on conic sections: the type, the standard form and every property, with the working.
• Finding the foci, vertices and directrices of a curve given in general form, including rotated ones.
• Writing the equation of a circle through three points, or of an ellipse or hyperbola from its foci and one more fact.
• Drawing a conic with its key points and lines for notes, worksheets or slides.
• Studying orbits, reflectors and lenses, whose shapes are ellipses, parabolas and hyperbolas.

Which conic is it?

For Ax² + Bxy + Cy² + Dx + Ey + F = 0, the discriminant B² − 4AC decides the family:

• B² − 4AC < 0: an ellipse (a circle when B = 0 and A = C);
• B² − 4AC = 0: a parabola;
• B² − 4AC > 0: a hyperbola (a rectangular one when A + C = 0).

The determinant Δ of the matrix [[A, B/2, D/2], [B/2, C, E/2], [D/2, E/2, F]] tells whether the curve is degenerate: when Δ = 0 it is a single point, two crossing lines, two parallel lines, one line, or no real points. An ellipse is real only when Δ and A + C have opposite signs; otherwise no real point satisfies the equation. These are the tests in OpenStax Precalculus 2e, “Rotation of Axes” and the classification of quadratic curves in MathWorld.

Standard forms and properties

• Circle (x − h)² + (y − k)² = r²: centre (h, k), radius r.
• Ellipse (x − h)²/a² + (y − k)²/b² = 1 with a > b: vertices (h ± a, k), co-vertices (h, k ± b), foci (h ± c, k) with c² = a² − b², eccentricity e = c/a < 1, directrices x = h ± a/e, latus rectum 2b²/a.
• Hyperbola (x − h)²/a² − (y − k)²/b² = 1: vertices (h ± a, k), foci (h ± c, k) with c² = a² + b², e = c/a > 1, asymptotes y − k = ±(b/a)(x − h), directrices x = h ± a/e, latus rectum 2b²/a.
• Parabola (x − h)² = 4p(y − k): vertex (h, k), focus (h, k + p), directrix y = k − p, axis x = h, latus rectum |4p|, e = 1.

With y in place of x the axes swap. These are the forms of NCERT Class 11 Mathematics, chapter 10 (Conic Sections). Every conic except the circle is also the set of points whose distance to a focus is e times their distance to a directrix; the builder uses this to find a parabola from a focus and a directrix.

Rotated conics: the xy-term

An xy-term means the curve’s axes are turned. Rotating the coordinate axes by the angle θ with cot 2θ = (A − C)/B (0° < θ < 90°) and substituting x = x′cos θ − y′sin θ, y = x′sin θ + y′cos θ removes it. The new coefficients of x′² and y′² are (A + C ± R)/2 with R = √((A − C)² + B²), and the x′- and y′-terms are D cos θ + E sin θ and E cos θ − D sin θ. The calculator then completes the square in x′ and y′ and turns every point and line back into x and y.

cos θ and sin θ follow from cos 2θ by the half-angle formulas, so they are exact when cos 2θ is a fraction (√3/2 and 1/2 for θ = 30°, 2√5/5 and √5/5 for cot 2θ = 3/4). For other angles, such as θ = 22.5°, the values that depend on the turn are given as decimals.

Building an equation from properties

• Circle: from the centre and the radius or a point on it, the ends of a diameter (the centre is their midpoint), or three points (solving x² + y² + Dx + Ey + F = 0 through each).
• Parabola: from the vertex and the focus, a focus and a directrix, the vertex and the directrix (the focus is the vertex’s mirror image away from the directrix), or the vertex, the direction of the axis and one point.
• Ellipse and hyperbola: from the centre and the semi-axes, from the two foci and one more fact (an axis length, the eccentricity, the latus rectum or a point: an ellipse’s distances to the foci add up to 2a, a hyperbola’s differ by 2a), from the two vertices and one more fact, or from the centre and two points when the axes are parallel to x and y.

The foci or vertices may lie anywhere, so the result can be a rotated conic; it is then analysed like a typed equation.

Limitations

• Coefficients are numbers, fractions, decimals and square roots (√ or sqrt); other functions such as sin, and letters other than x and y, are not accepted.
• Equations of degree 2 in x and y only; curves of higher degree and three-dimensional quadric surfaces are not covered.
• Exact values cover fractions and square roots such as 2 + √5 or (√6 + √2)/4; when the turn of a rotated conic has no such cosine, the values that depend on it are decimals with six significant digits.
• The graph is drawn to fit the curve’s key points with equal scales on both axes; read exact coordinates from the table.
• The centre-and-two-points builder assumes axes parallel to x and y.

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