$m=\frac{b}{a}$

$2\alpha$

eje focal

A = (-3.84, 5.82)
B = (9.88, 5.9)
a = Ray(A, B)
C = Point(a)
b = Segment(A, C)
D = (-3.14, 2.84)
E = (-1.34, 2.82)
c = Segment(D, E)
E' = Rotate(E, atan(b), D)
d = Segment(D, E')
α = Angle(E, D, E')
P = (9.43620225464765, 3.4006568772481476)
H_1 = (5.986240147384509, 4.362724862341246)
H_2 = (5.522371016659149, 1.6997724452141858)
F = (0.9867617384556353, 5.513807520067136)
G = (7.19940907754168, -1.546603326233281)
δ = Line(F, G)
f = Segment(H_1, H_2)
g = LineBisector(f)
H = Rotate(H_1, -2 * α, H_2)
h = Segment(H, H_2)
β = Angle(H, H_2, H_1)
i = OrthogonalLine(H_2, h)
I = Intersect(g, i)
J = Mirror(H_1, I)
k = CircumcircleArc(H_1, J, H_2)
K, L = Intersect(k, δ)
h_1 = Line(L, H_1)
h_2 = Line(L, H_2)
m = Line(P, h_1)
n = Line(P, h_2)
M = Intersect(n, h_1)
N = Intersect(m, h_2)
p = AngularBisector(M, L, N)
q = Line(P, p)
r = Line(P, p)
O = Intersect(h_1, r)
Q = Intersect(r, h_2)
s = Semicircle(P, O)
t = Semicircle(P, Q)
a_1 = OrthogonalLine(O, p)
b_1 = OrthogonalLine(Q, p)
R = Intersect(t, a_1)
c_1 = Circle(L, Segment[P, R])
S, T = Intersect(c_1, p)
d_1 = OrthogonalLine(S, p)
U = Intersect(h_2, d_1)
e_1 = Circle(L, U)
V, W = Intersect(e_1, p)
f_1 = Hyperbola(W, V, P)
Z = Intersect(s, b_1)
g_1 = Circle(L, Segment[P, Z])
A_1, B_1 = Intersect(g_1, p)
h_3 = OrthogonalLine(B_1, p)
C_1 = Intersect(h_3, h_2)
k_1 = Circle(L, C_1)
D_1, E_1 = Intersect(k_1, p)
p_1 = Hyperbola(E_1, D_1, P)

Tool: Hiperbola Punto Dos Asintotas (command HiperbolaPuntoDosAsintotas)
Dar un punto y las dos asíntotas
Input: P, h_1, h_2
Output: p, P', P'_1, P_1, P_2

E = Intersect(h_1, h_2)
P' = Mirror(P, E)
b = Line(P, h_2)
c = Line(P', h_1)
d = Line(P, h_1)
e = Line(P', h_2)
F = Intersect(e, d)
G = Midpoint(P', F)
f = Line(P, G)
H = Intersect(b, c)
g = Line(H, G)
h = Line(P', g)
P_1 = Intersect(f, h)
P'_1 = Mirror(P_1, E)
J = Midpoint(G, F)
i = Line(P, J)
j = Line(H, i)
K = Intersect(h_2, j)
k = Line(K, P')
P_2 = Intersect(i, k)
p = Conic(P', P_1, P_2, P, P'_1)