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Foci of Hyperbola

Results

${(-sqrt(a^2 + b^2), 0), (sqrt(a^2 + b^2), 0)}$

(-sqrt(a^2 + b^2) | 0
sqrt(a^2 + b^2) | 0)

Example plots

Equations

x(t) = a sec(t)
y(t) = b tan(t)

x^2/a^2 - y^2/b^2 = 1

r(θ) = (a b)/sqrt(b^2 cos^2(θ) - a^2 sin^2(θ))

(for a hyperbola with center at the origin, semimajor axis a parallel to the x-axis, and semiminor axis b parallel to the y-axis)

Conic properties

e = sqrt(b^2/a^2 + 1)

p = b^2/sqrt(a^2 + b^2)

L = b^2/a

y = -(b x)/a ∨ y = (b x)/a

x = -a^2/sqrt(a^2 + b^2) ∨ x = a^2/sqrt(a^2 + b^2)

Characteristic polynomial

λ sqrt(a^2 + b^2) + λ^2

Eigenvalues

λ_1 = -sqrt(a^2 + b^2)

λ_2 = 0

Eigenvectors

v_1 = (-1, 1)

v_2 = (0, 1)

Diagonalization

(-sqrt(a^2 + b^2) | 0
sqrt(a^2 + b^2) | 0) = S.J.S^(-1)
where
S = (0 | -1
1 | 1)
J = (0 | 0
0 | -sqrt(a^2 + b^2))
S^(-1) = (1 | 1
-1 | 0)