Hyperbola calculator

What is the equation of a hyperbola?

Centered at (h, k) with the transverse axis along x: (x − h)² ÷ a² − (y − k)² ÷ b² = 1. With the transverse axis along y, the y term comes first: (y − k)² ÷ a² − (x − h)² ÷ b² = 1. The distance from the center to each focus is c, where c² = a² + b². The term with the plus sign tells you which way the two branches open.

Example: (x − 2)²/36 − (y + 5)²/81 = 1

The center is (2, −5), a = 6 and b = 9, so c = √(36 + 81) = √117 = 3√13 ≈ 10.817. The vertices are (−4, −5) and (8, −5), the co-vertices (2, −14) and (2, 4), the foci (2 ± 3√13, −5), and the asymptotes y = ±1.5(x − 2) − 5. This is Example 5 in OpenStax Precalculus 2e, section 10.2.

How do I find the vertices, foci and asymptotes?

For a horizontal hyperbola the vertices are (h ± a, k), the co-vertices (h, k ± b), the foci (h ± c, k), and the asymptotes y = k ± (b ÷ a)(x − h). For a vertical one the vertices are (h, k ± a), the foci (h, k ± c), and the asymptotes y = k ± (a ÷ b)(x − h). For y²/64 − x²/36 = 1: a = 8, b = 6, c = 10, vertices (0, ±8), foci (0, ±10), asymptotes y = ±(4/3)x (OpenStax Example 4).

How do I turn Ax² + Cy² + Dx + Ey + F = 0 into standard form?

Complete the square in x and in y, move the constant to the right and divide so the right side is 1. For 9x² − 4y² − 36x − 40y − 388 = 0: 9(x − 2)² − 4(y + 5)² = 324, then dividing by 324 gives (x − 2)²/36 − (y + 5)²/81 = 1. A and C must have opposite signs; if the right side comes out as 0, the graph is just two crossing lines.

What is the eccentricity of a hyperbola?

e = c ÷ a, and it is always greater than 1 (OpenStax Calculus Volume 2, section 7.5). The bigger it is, the wider the branches open. For y²/49 − x²/32 = 1, c = 9 and a = 7, so e = 9/7 ≈ 1.286 (a and c are from OpenStax Example 1).

How is this different from the ellipse, parabola and circle calculators?

All four are conic sections. An ellipse adds its two distances to the foci, a hyperbola subtracts them, so a hyperbola has two separate branches and c² = a² + b² where an ellipse has c² = a² − b². For the other curves see the ellipse, parabola and circle equation calculators.

Where do the formulas come from?

From OpenStax Precalculus 2e, section 10.2 “The Hyperbola” (the standard forms and the table of vertices, foci and asymptotes) and OpenStax Intermediate Algebra 2e, section 11.4. The eccentricity e = c ÷ a comes from OpenStax Calculus Volume 2, section 7.5 “Conic Sections”. Their worked examples are part of this calculator’s automated tests.

Sources: OpenStax Precalculus 2e, section 10.2; OpenStax Intermediate Algebra 2e, section 11.4; OpenStax Calculus Volume 2, section 7.5. The other conic sections: ellipse, parabola and circle equation calculators.