\textrm{Graf nepřímé úměrnosti - \textbf{hyperbola}}

\textrm{Př. Lán pole zorá 6 traktorist\!ů za 1 hodinu.}\
\hspace{1cm}\textrm{Zobrazte body na grafu, které odpovídaj\!í situaci,}\
\hspace{1cm}\textrm{kdy budou lán orat 3 traktoristé, 2 traktoristé nebo 1 traktorista.}

x

y

y=\frac{6}{x}

1

2

3

6

1

2

3

6

\textrm{Grafem nepř\!ímé úměrnosti je jedna větev rovnoosé hyperboly}.\
\textrm{Hyperbola se středem v počátku, jejíž hlavní osou je osa\,}x\textrm{, je dána rovnicí:}\
\hspace{4cm}\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\
\textrm{Otočen\!ím souřadnicového systému o 45° lze funkci }y=\frac{6}{x}\textrm{ do tohoto tvaru p\!řepsat},\
\textrm{č\!ímž dokážeme, že grafem funkce je tedy skutečně hyperbola.}

\overline{x}

\overline{y}

x

x

x\sqrt2

\overline{y}

\overline{y}

\overline{y}\sqrt{2}

\textrm{Vidíme, že platí následující vztahy:}\
\overline{x}=x\sqrt{2}+\overline{y}\
y=x+\overline{y}\sqrt{2}\
\textrm{Odtud:}\
\vspace{1cm}x=\frac{\overline{x}-\overline{y}}{\sqrt{2}}\
\vspace{1.2cm}y=\frac{\overline{x}-\overline{y}}{\sqrt{2}}+\overline{y}\sqrt{2}=\frac{\overline{x}+\overline{y}}{\sqrt{2}}\
\vspace{1cm}\textrm{Dosadíme do rovnice: }x\cdot y=6\
\left( \frac{\overline{x}-\overline{y}}{\sqrt{2}}\right)\cdot\left(\frac{\overline{x}+\overline{y}}{\sqrt{2}}\right)=6\
\vspace{1cm}\frac{\overline{x}^2-\overline{y}^2}{12}=1\textrm{  ,tedy }\frac{\overline{x}^2}{12}-\frac{\overline{y}^2}{12}=1\dots\textrm{je rovnice hyperboly}\
\vspace{1cm}\textrm{Délka poloos:  }a=b=\sqrt{12}=2\sqrt{3}

\textrm{Odvození rovnice hyperboly ze vztahu } |EM|-|FM|=2a \textrm{ (Předpokládejme, že } |EM|>|FM|.\,)\
|EM|=2a+|FM|\,/^2\
|EM|^2=4a^2+4a\cdot |FM|+|FM|^2\
4xe=4a^2+4a\cdot|FM|\
\frac{xe}{a}-a=|FM|\,/^2\
\frac{x^2e^2}{a^2}-2xe+a^2=x^2-2xe+e^2+y^2\
\frac{x^2\cdot(\overbrace{e^2-a^2}^{b^{\,2}})}{a^2}-y^2=\overbrace{e^2-a^2}^{b^{\,2}}\,/\!:b^2\
\frac{x^2}{a^2}-\frac{y^2}{b^2}=1

E\,[-e,0]

S\,[0,0]

F\,[e,0]

M\,[x,y]

2a

o_1

|EM|=\sqrt{(x+e)^2+y^2}\ \Rightarrow\ |EM|^2=x^2+2xe+e^2+y^2\ 
|FM|=\sqrt{(x-e)^2+y^2}\ \Rightarrow\ |FM|^2=x^2-2xe+e^2+y^2\
\textrm{u hyperboly: }e^2=a^2+b^2

\textrm{Zdůvodnění, že hlavní vrcholy }A\textrm{ a }B\textrm{ leží na hyperbole,}\
\textrm{přičemž }A,B\in o_1,\ |AS|=|SB|=a.\
|\underbrace{\!|AE|\ }_{|SE|-|AS|}-\underbrace{\!|AF|\ }_{|AS|+|SF|}|=|\,|SE|-|AS|-|AS|-|SF|\,|=|e-a-a-e|=|\!-2a|=2a

E

A

o_1

S

B

F

a

e

O = Intersect(xAxis, yAxis)
X_1: (-8, 0)
X_2: (8, 0)
Y_1: (0, -8)
Y_2: (0, 8)
u = Vector(X_1, X_2)
v = Vector(Y_1, Y_2)
a = Segment(X_1, X_2)
b = Segment(Y_1, Y_2)
A: (1, 6)
B: (2, 3)
C: (3, 2)
D: (6, 1)
A' = Mirror(A, O)
B' = Mirror(B, O)
C' = Mirror(C, O)
D' = Mirror(D, O)
M_1: (0.8, 0)
M_2: (7, 0)
c = Conic(A, B, C, D, D')
d = Segment(M_1, M_2)
H_1 = Point(d)
e = OrthogonalLine(H_1, d)
H_2 = Intersect(c, e)
množina1 = Locus(H_2, H_1)
M_1' = Mirror(M_1, b)
M_2' = Mirror(M_2, b)
f = Segment(M_2', M_1')
G_1 = Point(f)
g = OrthogonalLine(G_1, f)
G_2, H = Intersect(c, g)
množina2 = Locus(G_2, G_1)
h = OrthogonalLine(A, u)
i = OrthogonalLine(B, u)
j = OrthogonalLine(C, u)
k = OrthogonalLine(D, u)
l = OrthogonalLine(A, v)
m = OrthogonalLine(B, v)
n = OrthogonalLine(C, v)
p = OrthogonalLine(D, v)
D_2 = Intersect(p, b)
C_2 = Intersect(n, b)
B_2 = Intersect(m, b)
A_2 = Intersect(l, b)
A_1 = Intersect(a, h)
B_1 = Intersect(a, i)
C_1 = Intersect(a, j)
D_1 = Intersect(a, k)
q = Segment(A_2, A)
r = Segment(B, B_2)
s = Segment(C, C_2)
t = Segment(D, D_2)
a_1 = Segment(A, A_1)
b_1 = Segment(B, B_1)
c_1 = Segment(C, C_1)
d_1 = Segment(D, D_1)
X_3 = Rotate(X_2, 45°, O)
X_4 = Mirror(X_3, O)
e_1 = Segment(X_4, X_3)
w = Vector(X_4, X_3)
Y_3 = Rotate(X_2, 135°, O)
Y_4 = Mirror(Y_3, O)
u_1 = Vector(Y_4, Y_3)
f_1 = Segment(Y_3, Y_4)
g_1 = OrthogonalLine(A, e_1)
h_1 = OrthogonalLine(A, f_1)
A_1' = Intersect(e_1, g_1)
A_2' = Intersect(h_1, f_1)
i_1 = Segment(A, A_1')
j_1 = Segment(A, A_2')
O' = Intersect(a_1, t)
mnohoúhelník1, o', o, a_2 = Polygon(O, A_1, O')
mnohoúhelník2, a_3, o'_1, a_1' = Polygon(O', A_1', A)
S_2' = Point(t)
k_1 = OrthogonalLine(S_2', u)
S_2 = Intersect(k_1, a)
S_1' = Point(h)
l_1 = OrthogonalLine(S_1', v)
S_1 = Intersect(l_1, b)
S_1'' = Point(h)
v_1 = Vector(S_1, S_1')
w_1 = Vector(S_1', S_1)
m_1 = Segment(A_1, S_1'')
n_1 = Vector(S_2, S_2')
p_1 = Vector(S_2', S_2)
q_1 = OrthogonalLine(O', e_1)
S_3 = Point(f_1)
r_1 = OrthogonalLine(S_3, f_1)
S_3' = Intersect(r_1, q_1)
S_3'' = Point(q_1)
s_1 = Vector(S_3, S_3')
t_1 = Vector(S_3', S_3)
b_2 = Segment(O', S_3'')
S_4 = Point(h_1)
c_2 = OrthogonalLine(S_4, e_1)
S_4' = Intersect(e_1, c_2)
S_4'' = Point(h_1)
d_2 = Segment(A, S_4'')
u_2 = Vector(S_4, S_4')
v_2 = Vector(S_4', S_4)
S_5 = Point(q_1)
S_5'' = Point(q_1)
e_2 = OrthogonalLine(S_5, q_1)
f_2 = OrthogonalLine(S_5'', q_1)
S_5' = Intersect(g_1, e_2)
S_5''' = Intersect(f_2, g_1)
w_2 = Vector(S_5, S_5')
g_2 = Vector(S_5', S_5)
h_2 = Segment(O', S_5'')
i_2 = Segment(A_1', S_5''')
S_6 = Point(q)
j_2 = OrthogonalLine(S_6, u)
S_6' = Intersect(t, j_2)
k_2 = Vector(S_6', S_6)
l_2 = Vector(S_6, S_6')
E = (-13.900375939849637, -1.3533834586466233)
m_2 = OrthogonalLine(E, v)
F = Point(m_2)
M = (-9.868421052631588, 1.381578947368415)
Z_1 = Point(m_2)
S = Midpoint(E, F)
Z_2 = Mirror(Z_1, S)
n_2 = Segment(Z_1, Z_2)
p_2 = Segment(E, M)
q_2 = Segment(F, M)
r_2 = Circle(M, F)
N = Intersect(r_2, p_2)
s_2 = Arc(r_2, N, F)
t_2 = OrthogonalLine(E, p_2)
b_3 = OrthogonalLine(N, p_2)
S_7 = Point(t_2)
S_7'' = Point(t_2)
c_3 = OrthogonalLine(S_7'', t_2)
d_3 = OrthogonalLine(S_7, t_2)
S_7''' = Intersect(c_3, b_3)
S_7' = Intersect(d_3, b_3)
e_3 = Segment(S_7'', E)
f_3 = Segment(S_7''', N)
u_3 = Vector(S_7, S_7')
v_3 = Vector(S_7', S_7)
E' = (-13.167293233082718, -6.484962406015046)
g_3 = Line(E', u)
S' = Point(g_3)
F' = Mirror(E', S')
A_3 = Point(g_3)
B_3 = Mirror(A_3, S')
Z_1' = Point(g_3)
Z_2' = Mirror(Z_1', S')
h_3 = Segment(Z_1', Z_2')
i_3 = OrthogonalLine(A_3, h_3)
j_3 = OrthogonalLine(S', h_3)
S_8 = Point(i_3)
S_8'' = Point(i_3)
k_3 = OrthogonalLine(S_8'', i_3)
l_3 = OrthogonalLine(S_8, i_3)
S_8''' = Intersect(k_3, j_3)
S_8' = Intersect(l_3, j_3)
m_3 = Segment(S_8'', A_3)
n_3 = Segment(S_8''', S')
w_3 = Vector(S_8, S_8')
p_3 = Vector(S_8', S_8)
S_9' = Mirror(S_8', h_3)
S_9''' = Mirror(S_8''', h_3)
q_3 = OrthogonalLine(E', h_3)
r_3 = OrthogonalLine(S_9', q_3)
s_3 = OrthogonalLine(S_9''', q_3)
S_9'' = Intersect(q_3, s_3)
S_9 = Intersect(r_3, q_3)
t_3 = Segment(E', S_9'')
a_4 = Segment(S', S_9''')
u_4 = Vector(S_9, S_9')
v_4 = Vector(S_9', S_9)