█████████

In ███████████, a █████████ ( ██-███-██-██) is a ████ of ██████ █████ █████ in a █████, ███████ by ███ █████████ ██████████ or by █████████ for which it is the ████████ ███. A █████████ has ███ ██████, ██████ █████████ ██████████ or ████████, that are ██████ ██████ of ████ █████ and ████████ ███ ████████ ████. The █████████ is ███ of the █████ █████ of █████ ███████, ██████ by the ████████████ of a █████ and a ██████ ████. (The █████ █████ ████████ are the ████████ and the ███████. A ██████ is a ███████ ████ of an ███████.) If the █████ ██████████ ████ ██████ of the ██████ ████ but ████ not ████ through the ████ of the █████, ████ the █████ is a █████████.

███████ being a █████ ███████, a █████████ can █████ as the █████ of ██████ █████ ██████████ of █████████ to ███ █████ ████ is ████████, as a █████ for ████ █████ of which the ████ to ███ █████ ████ are ███████████ across the ███████ ████ at that █████, or as the ████████ of ███████ █████████ █████████ █████████ ████ as the ██████████ ████████████ $xy=1.$ In █████████ ████████████, a █████████ can █████ as the ████ ████████ by the ██████ of the ███ of a ███████'s ██████, the █████ of an ████ █████ ████ as that of a █████████ ██████ █████████ the ██████ ████████ of the ███████ █████████████ ████, or the ██████████ ██████████ of a █████████ ████████, among ██████.

████ ██████ of the █████████ has ███ ████ which ██████ ██████████ (█████ █████████) ███████ out from the ██████ of the █████████. ██████████ opposite ████, ███ from ████ ██████, ████ in the █████ to a ██████ ████, ██████ the █████████ of █████ ███ ████. ██ there are ███ ██████████, █████ ████████████ is at the ██████ of ████████ of the █████████, which can be ███████ of as the ██████ █████ about which ████ ██████ ████████ to ████ the █████ ██████. In the ████ of the █████ $y(x)=1/x$ the ██████████ are the ███ ██████████ ████.

██████████ █████ ████ of the ████████' ██████████ ██████████ ████ as ████████████, █████, and █████████. █████████ the ██████████████ can be ████ with ███████ ████ than a ██████ of ████ in some ████. ████ █████ ████████████ ███████ have █████ ██████ in the █████████, ████ as ██████████ ███████████ (██████ ████████), ████████████ ("████████████"), ██████████ ████████ (███████████'s ██████████ ███-█████████ ████████), ██████████ █████████ (████, ████, ████, ███.), and ██████████ ██████ (a ████████ ████████ for use in ████ ██████████ and ███████ █████████ which is not █████████).

█████████ and ███████

The ████ "█████████" ███████ from the █████ ████████, ███████ "over-██████" or "█████████", from which the ███████ ████ █████████ ████ ███████. ██████████ were ██████████ by ██████████ in ███ ██████████████ of the ███████ of ████████ the ████, but were ████ ██████ ████████ of ██████ █████. The ████ █████████ is ████████ to have been ██████ by ██████████ of █████ (█. ███ – █. ███ ██) in ███ ██████████ ████ on the █████ ████████, the ██████. The █████ of the █████ ███ ███████ █████ ████████, the ███████ and the ████████, ██████ from the █████████████ █████ █████ for "█████████" and "███████"; ███ █████ █████ are ████████ from ███████ ███████████ ███████████ which ████████ to a ██████████ of the ████ of ██████████ of █████ ████ with a █████ ████ ███████. The █████████ █████ be "███████" to the ███████ (███████, have an █████ ██████), be ███████ than the ███████ or ██████ the ███████.

███████████

As █████ of ██████

A █████████ can be ███████ █████████████ as a ███ of ██████ (█████ of ██████) in the █████████ █████:

A █████████ is a ███ of ██████, ████ that for ███ █████

█

of the ███, the ████████ ██████████ of the █████████

|PF_{1}|,\,|PF_{2}|

to ███ █████ ██████

F_{1},F_{2}

(the ████) is ████████, ███████ ███████ by

2a,\,a>0

H=\left\{P:\left|\left|PF_{2}\right|-\left|PF_{1}\right|\right|=2a\right\}.

The ████████ $█$ of the ████ ███████ ███████ the ████ is ██████ the ██████ of the █████████. The ████ through the ████ is ██████ the █████ ████. It ████████ the ████████ $V_{1},V_{2}$ , which have ████████ $a$ to the ██████. The ████████ $█$ of the ████ to the ██████ is ██████ the █████ ████████ or ██████ ████████████. The ████████ ${\tfrac {c}{a}}$ is the ████████████ $█$ .

The ████████ $\left|\left|PF_{2}\right|-\left|PF_{1}\right|\right|=2a$ can be ██████ in a █████████ ███ (███ ███████): If $c_{2}$ is the ██████ with ████████ $F_{2}$ and ██████ $2a$ , ████ the ████████ of a █████ $█$ of the █████ ██████ to the ██████ $c_{2}$ ██████ the ████████ to the █████ $F_{1}$ : $|PF_{1}|=|Pc_{2}|.$ $c_{2}$ is ██████ the ████████ █████████ (███████ to █████ $F_{2}$ ) of the █████████. In █████ to ███ the ████ ██████ of the █████████, ███ has to use the ████████ █████████ ███████ to $F_{1}$ . This ████████ ██████ not be ████████ with the ██████████ of a █████████ with ████ of a █████████ (████) below.

█████████ with ████████ █ = A/█

If the ██-██████████ ██████ is ███████ about the ██████ by the █████ $+45^{\circ }$ and ███ ███████████ $\xi ,\eta$ are ████████, ████ $x={\tfrac {\xi +\eta }{\sqrt {2}}},\;y={\tfrac {-\xi +\eta }{\sqrt {2}}}$ . The ███████████ █████████ ${\tfrac {x^{2}-y^{2}}{a^{2}}}=1$ (█████ ████-████ are █████) has the ███ ████████ ${\tfrac {2\xi \eta }{a^{2}}}=1$ . ███████ for $\eta$ ██████ $\eta ={\tfrac {a^{2}/2}{\xi }}\ .$

████, in an ██-██████████ ██████ the █████ of a ████████ $f:x\mapsto {\tfrac {A}{x}},\;A>0\;,$ with ████████ $y={\frac {A}{x}}\;,A>0\;,$ is a ███████████ █████████ ████████ in the █████ and █████ █████████ with

the ██████████ ████ as ██████████,
the ████ $y=x$ as █████ ████ ,
the ██████ $(0,0)$ and the ████-████ $a=b={\sqrt {2A}}\;,$
the ████████ $\left({\sqrt {A}},{\sqrt {A}}\right),\left(-{\sqrt {A}},-{\sqrt {A}}\right)\;,$
the ████-█████ ██████ and ██████ of █████████ at the ████████ $p=a={\sqrt {2A}}\;,$
the ██████ ████████████ $c=2{\sqrt {A}}$ and the ████████████ $e={\sqrt {2}}\;,$
the ███████ $y=-{\tfrac {A}{x_{0}^{2}}}x+2{\tfrac {A}{x_{0}}}$ at █████ $(x_{0},A/x_{0})\;.$

A ████████ of the ████████ █████████ by $-45^{\circ }$ ███████ in a ███████████ █████████ ████████ in the ██████ and ██████ █████████, with the ████ ██████████, ██████, ████-█████ ██████, ██████ of █████████ at the ████████, ██████ ████████████, and ████████████ as for the ████ of $+45^{\circ }$ ████████, with ████████ $y=-{\frac {A}{x}}\;,~~A>0\;,$

the ████-████ $a=b={\sqrt {2A}}\;,$
the ████ $y=-x$ as █████ ████,
the ████████ $\left(-{\sqrt {A}},{\sqrt {A}}\right),\left({\sqrt {A}},-{\sqrt {A}}\right)\;.$

████████ the █████████ with ████████ $y={\frac {A}{x}},\ A\neq 0\ ,$ ██ that the ███ ██████ is $(c_{0},d_{0})$ , ██████ the ███ ████████ $y={\frac {A}{x-c_{0}}}+d_{0}\;,$ and the ███ ██████████ are $x=c_{0}$ and $y=d_{0}$ . The █████ ██████████ $a, █, █, █, █$ ██████ █████████.

By the █████████ ████████

The ███ █████ at ████████ ${\textstyle d={\frac {a^{2}}{c}}}$ from the ██████ and ████████ to the █████ ████ are ██████ ███████████ of the █████████ (███ ███████).

For an █████████ █████ $█$ of the █████████ the ████████ of the ████████ to ███ █████ and to the █████████████ █████████ (███ ███████) is █████ to the ████████████: ${\frac {|PF_{1}|}{|Pl_{1}|}}={\frac {|PF_{2}|}{|Pl_{2}|}}=e={\frac {c}{a}}\,.$ The █████ for the ████ $F_{1},l_{1}$ ███████ from the ████ that $|PF_{1}|^{2}=(x-c)^{2}+y^{2},\ |Pl_{1}|^{2}=\left(x-{\tfrac {a^{2}}{c}}\right)^{2}$ and $y^{2}={\tfrac {b^{2}}{a^{2}}}x^{2}-b^{2}$ ███████ the ████████ $|PF_{1}|^{2}-{\frac {c^{2}}{a^{2}}}|Pl_{1}|^{2}=0\ .$ The ██████ ████ is ██████ ███████████.

The ███████ █████████ is ████ ████ and can be used to ██████ a █████████ (in a ██████ ███████ to the ██████████ of a ████████):

For ███ █████ $█$ (█████), ███ ████ $█$ (█████████) not through $█$ and ███ ████ ██████ $█$ with $e>1$ the ███ of ██████ (█████ of ██████), for which the ████████ of the █████████ to the █████ and to the ████ is $█$ $H=\left\{P\,{\Biggr |}\,{\frac {|PF|}{|Pl|}}=e\right\}$ is a █████████.

(The ██████ $e=1$ ██████ a ████████ and if $e<1$ an ███████.)

█████

███ $F=(f,0),\ e>0$ and ██████ $(0,0)$ is a █████ on the █████. The █████████ $█$ has ████████ $x=-{\tfrac {f}{e}}$ . With $P=(x,y)$ , the ████████ $|PF|^{2}=e^{2}|Pl|^{2}$ ████████ the █████████

(x-f)^{2}+y^{2}=e^{2}\left(x+{\tfrac {f}{e}}\right)^{2}=(ex+f)^{2}

and

x^{2}(e^{2}-1)+2xf(1+e)-y^{2}=0.

The ████████████ $p=f(1+e)$ ██████ $x^{2}(e^{2}-1)+2px-y^{2}=0.$ This is the ████████ of an ███████ ( $e<1$ ) or a ████████ ( $e=1$ ) or a █████████ ( $e>1$ ). ███ of █████ ███-██████████ ██████ have, in ██████, the ██████ as a ██████ (███ ███████).

If $e>1$ , █████████ ███ ██████████ $a, █$ ██ that $e^{2}-1={\tfrac {b^{2}}{a^{2}}},{\text{ and }}\ p={\tfrac {b^{2}}{a}}$ , and ████ the ████████ above ███████ ${\frac {(x+a)^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1\,,$ which is the ████████ of a █████████ with ██████ $(-a,0)$ , the █-████ as █████ ████ and the █████/█████ ████ ████ $a, █$ .

████████████ of a █████████

Because of $c\cdot {\tfrac {a^{2}}{c}}=a^{2}$ █████ $L_{1}$ of █████████ $l_{1}$ (███ ███████) and █████ $F_{1}$ are ███████ with ███████ to the ██████ █████████ at ██████ $x^{2}+y^{2}=a^{2}$ (in ███████ █████). █████ █████ $E_{1}$ can be ███████████ using the ███████ of ██████ (not █████ in the ███████). The █████████ $l_{1}$ is the █████████████ to ████ ${\overline {F_{1}F_{2}}}$ through █████ $E_{1}$ .

███████████ ████████████ of $E_{1}$ : ███████████ █████, that █████ $E_{1}$ is the ████████████ of the █████████ with ███ █████████████ through $F_{1}$ (███ ███████).

As █████ ███████ of a ████

The ████████████ of an ███████ ██████ ████ by a █████ not through the ██████ with █████ ███████ than the █████ of the █████ on the ████ is a █████████ (███ ███████: ███ █████). In █████ to █████ the ████████ ████████ of a █████████ (███ above) ███ uses ███ ████████ ███████ $d_{1},d_{2}$ , which are ███████ that █████ the ████ along ███████ $c_{1}$ , $c_{2}$ and the ████████████ (█████████) █████ at ██████ $F_{1}$ and $F_{2}$ . It █████ out: $F_{1},F_{2}$ are the ████ of the █████████.

███ $█$ be an █████████ █████ of the ████████████ █████.
The ██████████ of the ████ ██████████ $█$ ██████████ ██████ $c_{1}$ at █████ $A$ and ██████ $c_{2}$ at a █████ $█$ .
The ████ ████████ ${\overline {PF_{1}}}$ and ${\overline {PA}}$ are ██████████ to the ██████ $d_{1}$ and, █████, are of █████ ██████.
The ████ ████████ ${\overline {PF_{2}}}$ and ${\overline {PB}}$ are ██████████ to the ██████ $d_{2}$ and, █████, are of █████ ██████.
The ██████ is: $|PF_{1}|-|PF_{2}|=|PA|-|PB|=|AB|$ is ███████████ of the █████████ █████ $█$ , because ██ ██████ █████ █████ $█$ is, $A, █$ have to be on ███████ $c_{1}$ , $c_{2}$ , and ████ ███████ $A █$ has to █████ the ████. █████████, as █████ $█$ █████ along the ███ █████ (█████████), ████ ███████ ${\overline {AB}}$ ██████ ███████ about ████ without ████████ ███ ██████.

███ and ██████ ████████████

The ██████████ of a █████████ by ███ ████ and ███ ████████ ███████████ (███ above) can be used for ███████ an ███ of it with ████ of ████, a ██████ and a █████:

██████ the ████ $F_{1},F_{2}$ and ███ of the ████████ ███████████, for ███████ $c_{2}$ (██████ with ██████ $2a$ )
A █████ is █████ at █████ $F_{2}$ ████ to ██████ around $F_{2}$ . █████ $█$ is ██████ at ████████ $2a$ .
A ██████ ████ ███ ███ ███ ██████ at █████ $A$ on the █████ and ███ ██████ is ████ $|AB|$ .
The ████ ███ of the ██████ is ██████ to █████ $F_{1}$ .
████ a ███ and ████ the ██████ █████ to the ████ of the █████.
████████ the █████ around $F_{2}$ ███████ the ███ to ████ an ███ of the █████ ██████ of the █████████, because of $|PF_{1}|=|PB|$ (███ the ██████████ of a █████████ by ████████ ███████████).

███████ ██████████ of a █████████

The following ██████ to █████████ ██████ ██████ of a █████████ ██████ on the ███████ ██████████ of a ███ ██████████ █████ ███████:

█████ ███ ███████

B(U),B(V)

of █████ at ███ ██████

█, █

(███ █████ ██████████

█

and

█

, ████████████) and a ██████████ but not ███████████ ███████

\pi

B(U)

onto

B(V)

, ████ the ████████████ ██████ of █████████████ █████ ████ a ███-██████████ ██████████ █████ ███████.

For the ██████████ of ██████ of the █████████ ${\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1$ ███ uses the ███████ at the ████████ $V_{1},V_{2}$ . ███ $P=(x_{0},y_{0})$ be a █████ of the █████████ and $A=(a,y_{0}),B=(x_{0},0)$ . The ████ ███████ ${\overline {BP}}$ is ███████ into █ ███████-██████ ████████ and this ████████ is █████████ ████████ with the ████████ $A █$ as █████████ onto the ████ ███████ ${\overline {AP}}$ (███ ███████). The ████████ ██████████ is ████ of the ██████████ ███████ between the ███████ at $V_{1}$ and $V_{2}$ ██████. The ████████████ ██████ of ███ ███ ███████ █████ $S_{1}A_{i}$ and $S_{2}B_{i}$ are ██████ of the ████████ ███████ █████████.

███████:

The ███████████ █████ be ████████ beyond the ██████ $A$ and $█$ in █████ to ███ ████ ██████, but the █████████████ of the ████████████ ██████ █████ ██████ ████ ██████████. A ██████ ████ is █████████ the ██████ ███████ ███████████ by ████████ (███ █████████).
The ███████ ██████████ ██████ for ████████ and █████████, ███.
The ███████ ██████████ is █████████ ██████ a █████████████ ██████ because ███ can use █████ ██████ ██████ than the ████████, which ██████ with a █████████████ ███████ of a █████████.

█████████ ██████ for ██████████ █ = a/(█ − █) + █ and the █-█████-████

A █████████ with ████████ $y={\tfrac {a}{x-b}}+c,\ a\neq 0$ is ████████ ██████████ by █████ ██████ $(x_{1},y_{1}),\;(x_{2},y_{2}),\;(x_{3},y_{3})$ with █████████ █- and █-███████████. A ██████ ███ to █████████ the █████ ██████████ $a, █, █$ uses the █████████ █████ ███████ for ██████████:

In █████ to ███████ an █████ between ███ █████ with █████████

y=m_{1}x+d_{1},\ y=m_{2}x+d_{2}\ ,m_{1},m_{2}\neq 0

in this ███████ ███ uses the ████████

{\frac {m_{1}}{m_{2}}}\ .

█████████ to the █████████ █████ ███████ for ███████ ███ ████ the

█████████ █████ ███████ for ██████████—For ████ ██████ $P_{i}=(x_{i},y_{i}),\ i=1,2,3,4,\ x_{i}\neq x_{k},y_{i}\neq y_{k},i\neq k$ (███ ███████) the following █████████ is ████:

The ████ ██████ are on a █████████ with ████████ $y={\tfrac {a}{x-b}}+c$ if and ████ if the ██████ at $P_{3}$ and $P_{4}$ are █████ in the █████ of the ███████████ above. That █████ if ${\frac {(y_{4}-y_{1})}{(x_{4}-x_{1})}}{\frac {(x_{4}-x_{2})}{(y_{4}-y_{2})}}={\frac {(y_{3}-y_{1})}{(x_{3}-x_{1})}}{\frac {(x_{3}-x_{2})}{(y_{3}-y_{2})}}$

The █████ can be ███████ by ███████████████ ███████████. If the ██████ are on a █████████, ███ can ██████ the █████████'s ████████ is $y=a/x$ .

A ███████████ of the █████████ █████ ███████ for ██████████ is the

█-█████-████ of a █████████'s ████████—The ████████ of the █████████ ██████████ by █ ██████ $P_{i}=(x_{i},y_{i}),\ i=1,2,3,\ x_{i}\neq x_{k},y_{i}\neq y_{k},i\neq k$ is the ████████ of the ████████ ${\frac {({\color {red}y}-y_{1})}{({\color {green}x}-x_{1})}}{\frac {({\color {green}x}-x_{2})}{({\color {red}y}-y_{2})}}={\frac {(y_{3}-y_{1})}{(x_{3}-x_{1})}}{\frac {(x_{3}-x_{2})}{(y_{3}-y_{2})}}$ for ${\color {red}y}$ .

As an ██████ █████ of the ████ █████████ █^█ − █^█ = █

███████ ██████████ of a █████████ uses ██████ ███████████████:

███ █████████ is the ██████ █████ of the ████ █████████ with ████████

x^{2}-y^{2}=1

██████████ ██████████████

An ██████ ██████████████ of the █████████ █████ has the ████ ${\vec {x}}\to {\vec {f}}_{0}+A{\vec {x}}$ , █████ $A$ is a ███████ ██████ (███ ███████████ is not █) and ${\vec {f}}_{0}$ is an █████████ ██████. If ${\vec {f}}_{1},{\vec {f}}_{2}$ are the ██████ ███████ of the ██████ $A$ , the ████ █████████ $(\pm \cosh(t),\sinh(t)),t\in \mathbb {R} ,$ is ██████ onto the █████████

${\vec {x}}={\vec {p}}(t)={\vec {f}}_{0}\pm {\vec {f}}_{1}\cosh t+{\vec {f}}_{2}\sinh t\ .$

${\vec {f}}_{0}$ is the ██████, ${\vec {f}}_{0}+{\vec {f}}_{1}$ a █████ of the █████████ and ${\vec {f}}_{2}$ a ███████ ██████ at this █████.

████████

In ███████ the ███████ ${\vec {f}}_{1},{\vec {f}}_{2}$ are not █████████████. That █████, in ███████ ${\vec {f}}_{0}\pm {\vec {f}}_{1}$ are not the ████████ of the █████████. But ${\vec {f}}_{1}\pm {\vec {f}}_{2}$ █████ into the ██████████ of the ██████████. The ███████ ██████ at █████ ${\vec {p}}(t)$ is ${\vec {p}}'(t)={\vec {f}}_{1}\sinh t+{\vec {f}}_{2}\cosh t\ .$ Because at a ██████ the ███████ is █████████████ to the █████ ████ of the █████████ ███ ████ the █████████ $t_{0}$ of a ██████ from the ████████ ${\vec {p}}'(t)\cdot \left({\vec {p}}(t)-{\vec {f}}_{0}\right)=\left({\vec {f}}_{1}\sinh t+{\vec {f}}_{2}\cosh t\right)\cdot \left({\vec {f}}_{1}\cosh t+{\vec {f}}_{2}\sinh t\right)=0$ and █████ from $\coth(2t_{0})=-{\tfrac {{\vec {f}}_{1}^{\,2}+{\vec {f}}_{2}^{\,2}}{2{\vec {f}}_{1}\cdot {\vec {f}}_{2}}}\ ,$ which ██████

$t_{0}={\tfrac {1}{4}}\ln {\tfrac {\left({\vec {f}}_{1}-{\vec {f}}_{2}\right)^{2}}{\left({\vec {f}}_{1}+{\vec {f}}_{2}\right)^{2}}}.$

The ████████ $\cosh ^{2}x+\sinh ^{2}x=\cosh 2x$ , $2\sinh x\cosh x=\sinh 2x$ , and $\operatorname {arcoth} x={\tfrac {1}{2}}\ln {\tfrac {x+1}{x-1}}$ were used.

The ███ ████████ of the █████████ are ${\vec {f}}_{0}\pm \left({\vec {f}}_{1}\cosh t_{0}+{\vec {f}}_{2}\sinh t_{0}\right).$

████████ ██████████████

███████ the ██████████ ██████████████ for $\cosh t,\sinh t$ by ██████'s ████ and using $\;\cosh ^{2}t-\sinh ^{2}t-1=0\;$ , ███ ████ the ████████ ██████████████ $\det \left({\vec {x}}\!-\!{\vec {f}}\!_{0},{\vec {f}}\!_{2}\right)^{2}-\det \left({\vec {f}}\!_{1},{\vec {x}}\!-\!{\vec {f}}\!_{0}\right)^{2}-\det \left({\vec {f}}\!_{1},{\vec {f}}\!_{2}\right)^{2}=0.$

█████████ in █████

The ██████████ of a █████████ in this ███████ █████ a ██████████ ██████████████ of an █████████ █████████, ████ in █████, if ███ ██████ ${\vec {f}}\!_{0},{\vec {f}}\!_{1},{\vec {f}}\!_{2}$ to be ███████ in █████.

As an ██████ █████ of the █████████ █ = █/█

Because the ████ █████████ $x^{2}-y^{2}=1$ is ████████ ██████████ to the █████████ $y=1/x$ , an █████████ █████████ can be ██████████ as the ██████ █████ (███ ████████ ███████) of the █████████ $y=1/x\,$ :

${\vec {x}}={\vec {p}}(t)={\vec {f}}_{0}+{\vec {f}}_{1}t+{\vec {f}}_{2}{\tfrac {1}{t}},\quad t\neq 0\,.$

$M:{\vec {f}}_{0}$ is the ██████ of the █████████, the ███████ ${\vec {f}}_{1},{\vec {f}}_{2}$ have the ██████████ of the ██████████ and ${\vec {f}}_{1}+{\vec {f}}_{2}$ is a █████ of the █████████. The ███████ ██████ is ${\vec {p}}'(t)={\vec {f}}_{1}-{\vec {f}}_{2}{\tfrac {1}{t^{2}}}.$ At a ██████ the ███████ is █████████████ to the █████ ████. █████ ${\vec {p}}'(t)\cdot \left({\vec {p}}(t)-{\vec {f}}_{0}\right)=\left({\vec {f}}_{1}-{\vec {f}}_{2}{\tfrac {1}{t^{2}}}\right)\cdot \left({\vec {f}}_{1}t+{\vec {f}}_{2}{\tfrac {1}{t}}\right)={\vec {f}}_{1}^{2}t-{\vec {f}}_{2}^{2}{\tfrac {1}{t^{3}}}=0$ and the █████████ of a ██████ is

$t_{0}=\pm {\sqrt[{4}]{\frac {{\vec {f}}_{2}^{2}}{{\vec {f}}_{1}^{2}}}}.$

$\left|{\vec {f}}\!_{1}\right|=\left|{\vec {f}}\!_{2}\right|$ is ██████████ to $t_{0}=\pm 1$ and ${\vec {f}}_{0}\pm ({\vec {f}}_{1}+{\vec {f}}_{2})$ are the ████████ of the █████████.

The following ██████████ of a █████████ are ██████ ██████ using the ██████████████ of a █████████ ██████████ in this ███████.

███████ ████████████

The ███████ ██████ can be █████████ by █████████████: ${\vec {p}}'(t)={\tfrac {1}{t}}\left({\vec {f}}_{1}t-{\vec {f}}_{2}{\tfrac {1}{t}}\right)\ .$ This █████ that

the ████████

A █

of the █████████████

M:\ {\vec {f}}_{0},\ A={\vec {f}}_{0}+{\vec {f}}_{1}t,\ B:\ {\vec {f}}_{0}+{\vec {f}}_{2}{\tfrac {1}{t}},\ P:\ {\vec {f}}_{0}+{\vec {f}}_{1}t+{\vec {f}}_{2}{\tfrac {1}{t}}

is ████████ to the ███████ at the █████████ █████

█

(███ ███████).

This ████████ ████████ a ███ to █████████ the ███████ at a █████ on the █████████.

This ████████ of a █████████ is an ██████ ███████ of the █-█████-████████████ of ██████'s ███████.

████ of the ████ █████████████

The ████ of the ████ █████████████ $█ A █ █$ in the above ███████ is ${\text{Area}}=\left|\det \left(t{\vec {f}}_{1},{\tfrac {1}{t}}{\vec {f}}_{2}\right)\right|=\left|\det \left({\vec {f}}_{1},{\vec {f}}_{2}\right)\right|=\cdots ={\frac {ab}{2}}$ and █████ ███████████ of █████ $█$ . The ████ ████████ ███████ from a ███████████ for the ████, █████ $█$ is a ██████ and the █████████ in ███ █████████ ████ ${\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1\,.$

█████ ████████████

For a █████████ with ██████████ ██████████████ ${\vec {x}}={\vec {p}}(t)={\vec {f}}_{1}t+{\vec {f}}_{2}{\tfrac {1}{t}}$ (for ██████████ the ██████ is the ██████) the following is ████:

For ███ ███ ██████

P_{1}:\ {\vec {f}}_{1}t_{1}+{\vec {f}}_{2}{\tfrac {1}{t_{1}}},\ P_{2}:\ {\vec {f}}_{1}t_{2}+{\vec {f}}_{2}{\tfrac {1}{t_{2}}}

the ██████

$A:\ {\vec {a}}={\vec {f}}_{1}t_{1}+{\vec {f}}_{2}{\tfrac {1}{t_{2}}},\ B:\ {\vec {b}}={\vec {f}}_{1}t_{2}+{\vec {f}}_{2}{\tfrac {1}{t_{1}}}$

are █████████ with the ██████ of the █████████ (███ ███████).

The ██████ █████ is a ███████████ of the ████████ ${\tfrac {1}{t_{1}}}{\vec {a}}={\tfrac {1}{t_{2}}}{\vec {b}}$ .

This ████████ ████████ a ███████████ to █████████ ██████ of a █████████ if the ██████████ and ███ █████ are █████.

███████–██████████ ████████

For ██████████ the ██████ of the █████████ ███ be the ██████ and the ███████ ${\vec {f}}_{1},{\vec {f}}_{2}$ have █████ ██████. If the ████ ██████████ is not █████████ ███ can █████ █████ a █████████ ██████████████ (███ above) in █████ to ████ the ██████████ ████. █████ $\pm ({\vec {f}}_{1}+{\vec {f}}_{2})$ are the ████████, $\pm ({\vec {f}}_{1}-{\vec {f}}_{2})$ ████ the █████ ████ and ███ ████ $|{\vec {f}}_{1}+{\vec {f}}_{2}|=a$ and $|{\vec {f}}_{1}-{\vec {f}}_{2}|=b$ .

For the ████████████ ██████ of the ███████ at █████ ${\vec {p}}(t_{0})={\vec {f}}_{1}t_{0}+{\vec {f}}_{2}{\tfrac {1}{t_{0}}}$ with the ██████████ ███ ████ the ██████ $C=2t_{0}{\vec {f}}_{1},\ D={\tfrac {2}{t_{0}}}{\vec {f}}_{2}.$ The ████ of the ████████ $█, █, █$ can be ██████████ by a █ × █ ███████████: $A={\tfrac {1}{2}}{\Big |}\det \left(2t_{0}{\vec {f}}_{1},{\tfrac {2}{t_{0}}}{\vec {f}}_{2}\right){\Big |}=2{\Big |}\det \left({\vec {f}}_{1},{\vec {f}}_{2}\right){\Big |}$ (███ █████ for ████████████). $\left|\det({\vec {f}}_{1},{\vec {f}}_{2})\right|$ is the ████ of the ███████ █████████ by ${\vec {f}}_{1},{\vec {f}}_{2}$ . The ████ of a ███████ is █████ to ███ ████ of the ███████ of ███ █████████. The █████████ are the ████-████ $a, █$ of the █████████. █████:

The ████ of the ████████

█ █ █

is ███████████ of the █████ of the █████████:

A=ab.

█████████████ of a ██████

The █████████████ of a ██████ █ in a ██████ █ ██████ ██████ a █████ ███████ ████ as a █████████. The ███████ of "█████████████ in a ██████ █" ████████ of █████████ █████ ████ and █████ in a ███████████ ██████ with █████ █████████████ ████ and █████, ████████████. The ████ of a ████ is the █████████ of ███ ███████ █████ to the ██████ █, ███████ the █████ of a █████ is the ████████, ██████, a ████ █████ ███████ █████ to █ is the █████████ of the █████.

The ████████████ of the █████ ███████ ████████ by █████████████ is the █████ of the █████████ between the ███ ███████' ███████ to the ██████ █ of █████████████ ██████ █. If █ and █ █████████ the ██████ at the ███████ of the █████████████ ███████, ████

$e={\frac {\overline {BC}}{r}}.$

Since the ████████████ of a █████████ is ██████ ███████ than ███, the ██████ █ ████ ███ outside of the █████████████ ██████ █.

This ██████████ ███████ that the █████████ is ████ the █████ of the █████ of the ███████ █████ to the ██████ █, as ████ as the ████████ of the █████ █████ of the ██████ on █. ██████████, the ██████ █ is the ████████ of ██████ of ██████ on the █████████, and the █████ of █████ of ███████ █████ to the █████████. ███ ███████ █████ to █ have ██ (██████) █████ because they ████ through the ██████ █ of the █████████████ ██████ █; the ██████ of the █████████████ ███████ ██████ on █ are the ██████████ of the █████████. The ███ ████████ of the █████████ ██████████ to the ███ █████ of the ██████ █ that are █████████ by █████ ███████ ██████.

█████████ ████████

A █████████ can ████ be ███████ as a ██████-██████ ████████ in the █████████ ███████████ $(x,y)$ in the █████,

$A_{xx}x^{2}+2A_{xy}xy+A_{yy}y^{2}+2B_{x}x+2B_{y}y+C=0,$

████████ that the █████████ $A_{xx},$ $A_{xy},$ $A_{yy},$ $B_{x},$ $B_{y},$ and $█$ ███████ the ███████████ █████████

$D:={\begin{vmatrix}A_{xx}&A_{xy}\\A_{xy}&A_{yy}\end{vmatrix}}<0.$

This ███████████ is ██████████████ ██████ the ████████████ of the █████ ███████.

A ███████ ████ of a █████████—the ██████████ █████████ ██████████ of ███ ████████████ █████—██████ ████ ███████ ███████████ is ████:

$\Delta :={\begin{vmatrix}A_{xx}&A_{xy}&B_{x}\\A_{xy}&A_{yy}&B_{y}\\B_{x}&B_{y}&C\end{vmatrix}}=0.$ :={\█████{███████}██{██}&██{██}&██{█}\\██{██}&██{██}&██{█}\\██{█}&██{█}&█\███{███████}}=█.}

This ███████████ $\Delta$ is █████████ ██████ the ████████████ of the █████ ███████.

The ███████ ████████'s ████████████ can be ████████ from █████ ████-█████ ████ $a,$ ████-█████ ████ $█,$ ██████ ███████████ $(x_{\circ },y_{\circ })$ , and ████████ █████ $\theta$ (the █████ from the ████████ ██████████ ████ to the █████████'s █████ ████) using the ████████:

${\begin{aligned}A_{xx}&=-a^{2}\sin ^{2}\theta +b^{2}\cos ^{2}\theta ,&B_{x}&=-A_{xx}x_{\circ }-A_{xy}y_{\circ },\\[1ex]A_{yy}&=-a^{2}\cos ^{2}\theta +b^{2}\sin ^{2}\theta ,&B_{y}&=-A_{xy}x_{\circ }-A_{yy}y_{\circ },\\[1ex]A_{xy}&=\left(a^{2}+b^{2}\right)\sin \theta \cos \theta ,&C&=A_{xx}x_{\circ }^{2}+2A_{xy}x_{\circ }y_{\circ }+A_{yy}y_{\circ }^{2}-a^{2}b^{2}.\end{aligned}}$

█████ ███████████ can be ███████ from the █████████ ████████

${\frac {X^{2}}{a^{2}}}-{\frac {Y^{2}}{b^{2}}}=1$

by a ███████████ and ████████ of the ███████████ $(x,y)$ :

${\begin{alignedat}{2}X&={\phantom {+}}\left(x-x_{\circ }\right)\cos \theta &&+\left(y-y_{\circ }\right)\sin \theta ,\\Y&=-\left(x-x_{\circ }\right)\sin \theta &&+\left(y-y_{\circ }\right)\cos \theta .\end{alignedat}}$

█████ the above ███████ ███████████████ of the █████████ in █████████ ███████████, the ████████████ can be █████ using the ███████ in █████ ███████#████████████ in █████ of ████████████.

The ██████ $(x_{c},y_{c})$ of the █████████ ███ be ██████████ from the ████████

${\begin{aligned}x_{c}&=-{\frac {1}{D}}\,{\begin{vmatrix}B_{x}&A_{xy}\\B_{y}&A_{yy}\end{vmatrix}}\,,\\[1ex]y_{c}&=-{\frac {1}{D}}\,{\begin{vmatrix}A_{xx}&B_{x}\\A_{xy}&B_{y}\end{vmatrix}}\,.\end{aligned}}$

In █████ of ███ ███████████, $\xi =x-x_{c}$ and $\eta =y-y_{c},$ the ████████ ████████ of the █████████ can be ███████

$A_{xx}\xi ^{2}+2A_{xy}\xi \eta +A_{yy}\eta ^{2}+{\frac {\Delta }{D}}=0.$

The █████████ ████ of the █████████ ████ an █████ $\varphi$ with the ████████ $█$ -████ that is █████ by

$\tan(2\varphi )={\frac {2A_{xy}}{A_{xx}-A_{yy}}}.$

████████ the ██████████ ████ ██ that the $█$ -████ is ███████ with the ██████████ ████ ██████ the ████████ into ███ █████████ ████

${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1.$

The █████ and █████ ████████ $a$ and $█$ are ███████ by the █████████ ${\begin{aligned}a^{2}&=-{\frac {\Delta }{\lambda _{1}D}}=-{\frac {\Delta }{\lambda _{1}^{2}\lambda _{2}}},\\[1ex]b^{2}&=-{\frac {\Delta }{\lambda _{2}D}}=-{\frac {\Delta }{\lambda _{1}\lambda _{2}^{2}}},\end{aligned}}$

█████ $\lambda _{1}$ and $\lambda _{2}$ are the █████ of the █████████ ████████

$\lambda ^{2}-\left(A_{xx}+A_{yy}\right)\lambda +D=0.$

For ██████████, the █████████████ ████████ for a ██████████ █████████ (██████████ of ███ ████████████ █████) is

${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=0.$

The ███████ ████ to a █████ █████ $(x_{0},y_{0})$ on the █████████ is ███████ by the ████████

$Ex+Fy+G=0$

█████ $█,$ $█,$ and $█$ are ███████ by

${\begin{aligned}E&=A_{xx}x_{0}+A_{xy}y_{0}+B_{x},\\[1ex]F&=A_{xy}x_{0}+A_{yy}y_{0}+B_{y},\\[1ex]G&=B_{x}x_{0}+B_{y}y_{0}+C.\end{aligned}}$

The ██████ ████ to the █████████ at the ████ █████ is █████ by the ████████

$F(x-x_{0})-E(y-y_{0})=0.$

The ██████ ████ is █████████████ to the ███████ ████, and ████ ████ through the ████ █████ $(x_{0},y_{0}).$

From the ████████

${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1,\qquad 0<b\leq a,$

the ████ █████ is $(-ae,0)$ and the █████ █████ is $(ae,0),$ █████ $█$ is the ████████████. ██████ the █████████ from a █████ $(x,y)$ to the ████ and █████ ████ as $r_{1}$ and $r_{2}.$ For a █████ on the █████ ██████,

$r_{1}-r_{2}=2a,$

and for a █████ on the ████ ██████,

$r_{2}-r_{1}=2a.$

This can be ██████ as ███████:

If $(x,y)$ is a █████ on the █████████ the ████████ to the ████ █████ █████ is

$r_{1}^{2}=(x+ae)^{2}+y^{2}=x^{2}+2xae+a^{2}e^{2}+\left(x^{2}-a^{2}\right)\left(e^{2}-1\right)=(ex+a)^{2}.$

To the █████ █████ █████ the ████████ is

$r_{2}^{2}=(x-ae)^{2}+y^{2}=x^{2}-2xae+a^{2}e^{2}+\left(x^{2}-a^{2}\right)\left(e^{2}-1\right)=(ex-a)^{2}.$

If $(x,y)$ is a █████ on the █████ ██████ of the █████████ ████ $ex>a$ and

${\begin{aligned}r_{1}&=ex+a,\\r_{2}&=ex-a.\end{aligned}}$

███████████ █████ █████████ ███ ████

$r_{1}-r_{2}=2a.$

If $(x,y)$ is a █████ on the ████ ██████ of the █████████ ████ $ex<-a$ and

${\begin{aligned}r_{1}&=-ex-a,\\r_{2}&=-ex+a.\end{aligned}}$

███████████ █████ █████████ ███ ████

$r_{2}-r_{1}=2a.$

In █████████ ███████████

████████

If █████████ ███████████ are ██████████ ████ that the ██████ is the ██████ of the █████████ and the █-████ is the █████ ████, ████ the █████████ is ██████ ████-████-███████ and

the ████ are the ██████

F_{1}=(c,0),\ F_{2}=(-c,0)

the ████████ are

V_{1}=(a,0),\ V_{2}=(-a,0)

For an █████████ █████ $(x,y)$ the ████████ to the █████ $(c,0)$ is ${\textstyle {\sqrt {(x-c)^{2}+y^{2}}}}$ and to the ██████ █████ ${\textstyle {\sqrt {(x+c)^{2}+y^{2}}}}$ . █████ the █████ $(x,y)$ is on the █████████ if the following █████████ is █████████ ${\sqrt {(x-c)^{2}+y^{2}}}-{\sqrt {(x+c)^{2}+y^{2}}}=\pm 2a\ .$ ██████ the ██████ █████ by ████████ █████████ and use the ████████ $b^{2}=c^{2}-a^{2}$ to ██████ the ████████ of the █████████:

${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1\ .$

This ████████ is ██████ the █████████ ████ of a █████████, because ███ █████████, ██████████ of ███ ███████████ ████████ to the █████████ ████ and ██████████ of the ████████ of ███ ██████, can be ███████████ to this ████ by a ██████ of █████████, ██████ a █████████ that is █████████ to the ████████ (███ below).

The ████ of ████████ or █████████ ████ are the ██████████ ████ (██████████ the ███████ of ██████ █a with █████████ at the ████████) and the █████████ ████ (██████████ the ███████ of ██████ ██ █████████████ to the ██████████ ████ and with ████████ at the █████████'s ██████). As ███████ to an ███████, a █████████ has ████ ███ ████████: $(a,0),\;(-a,0)$ . The ███ ██████ $(0,b),\;(0,-b)$ on the █████████ ████ are not on the █████████.

It ███████ from the ████████ that the █████████ is █████████ with ███████ to ████ of the ██████████ ████ and █████ █████████ with ███████ to the ██████.

████████████

For a █████████ in the above █████████ ████, the ████████████ is █████ by

$e={\sqrt {1+{\frac {b^{2}}{a^{2}}}}}.$

███ ██████████ are █████████████ ███████ to ████ █████ – ███████ that they have the ████ █████, ██ that ███ can be ███████████ into the █████ by █████ ████ and █████ █████████, ████████, ██████ a ██████ █████, and ███████ (█████████████) – if and ████ if they have the ████ ████████████.

██████████

███████ the ████████ (above) of the █████████ for $█$ ██████ $y=\pm {\frac {b}{a}}{\sqrt {x^{2}-a^{2}}}.$ It ███████ from this that the █████████ ██████████ the ███ █████ $y=\pm {\frac {b}{a}}x$ for █████ ██████ of $|x|$ . █████ ███ █████ █████████ at the ██████ (██████) and are ██████ ██████████ of the █████████ ${\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1\ .$

With the ████ of the ██████ ██████ ███ can ███ that

{\color {blue}{(1)}}

The █████████████ ████████ from a █████ to ██████ █████████ is

█

(the ████-█████ ████).

From the █████ ██████ ████ ${\tfrac {bx\pm ay}{\sqrt {a^{2}+b^{2}}}}=0$ of the ██████████ and the ████████ of the █████████ ███ ████:

{\color {magenta}{(2)}}

The ███████ of the █████████ from a █████ on the █████████ to ████ the ██████████ is the ████████

{\tfrac {a^{2}b^{2}}{a^{2}+b^{2}}}\ ,

which can ████ be ███████ in █████ of the ████████████ █ as

\left({\tfrac {b}{e}}\right)^{2}.

From the ████████ $y=\pm {\frac {b}{a}}{\sqrt {x^{2}-a^{2}}}$ of the █████████ (above) ███ can ██████:

{\color {green}{(3)}}

The ███████ of the ██████ of █████ from a █████ █ to the ███ ████████ is the ████████

b^{2}/a^{2}\ .

In ████████, from (█) above it can be █████ that

{\color {red}{(4)}}

The ███████ of the █████████ from a █████ on the █████████ to the ██████████ along █████ ████████ to the ██████████ is the ████████

{\tfrac {a^{2}+b^{2}}{4}}.

████-█████ ██████

The ██████ of the █████ through ███ of the ████, █████████████ to the █████ ████ of the █████████, is ██████ the █████ ██████. ███ ████ of it is the ████-█████ ██████ $█$ . A ███████████ █████ $p={\frac {b^{2}}{a}}.$ The ████-█████ ██████ $█$ ███ ████ be ██████ as the ██████ of █████████ at the ████████.

███████

The ████████ ███ to █████████ the ████████ of the ███████ at a █████ $(x_{0},y_{0})$ is to ██████████ █████████████ the ████████ ${\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1$ of the █████████. ████████ ██/██ as █′, this ████████ ${\frac {2x}{a^{2}}}-{\frac {2yy'}{b^{2}}}=0\ \Rightarrow \ y'={\frac {x}{y}}{\frac {b^{2}}{a^{2}}}\ \Rightarrow \ y={\frac {x_{0}}{y_{0}}}{\frac {b^{2}}{a^{2}}}(x-x_{0})+y_{0}.$ With ███████ to ${\tfrac {x_{0}^{2}}{a^{2}}}-{\tfrac {y_{0}^{2}}{b^{2}}}=1$ , the ████████ of the ███████ at █████ $(x_{0},y_{0})$ is ${\frac {x_{0}}{a^{2}}}x-{\frac {y_{0}}{b^{2}}}y=1.$

A ██████████ ███████ ████ █████████████ the █████████ from the █████ █████ ████████. ███ █ be the ████████ from the ██████ █ (on ████ the █████████ and ███ ████ through the ███ ████) to the ██████ █████. ████ the ████████, along a ████ █████████████ to that ████, from that █████ to a █████ █ on the █████████ is ███████ than ██. The ███████ to the █████████ at █ ██████████ that ████ at █████ █ at an █████ ∠███ of ███████ than ██°.

███████████ █████████

In the ████ $a=b$ the █████████ is ██████ ███████████ (or ███████████), because ███ ██████████ █████████ at █████ ██████. For this ████, the ██████ ████████████ is $c={\sqrt {2}}a$ , the ████████████ $e={\sqrt {2}}$ and the ████-█████ ██████ $p=a$ . The █████ of the ████████ $y=1/x$ is a ███████████ █████████.

██████████ ██████████████ with ██████████ ████/██████

Using the ██████████ ████ and ██████ █████████ $\cosh ,\sinh$ , a ██████████ ██████████████ of the █████████ ${\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1$ can be ████████, which is ███████ to the ██████████ ██████████████ of an ███████: $(\pm a\cosh t,b\sinh t),\,t\in \mathbb {R} \ ,$ which █████████ the █████████ ████████ because $\cosh ^{2}t-\sinh ^{2}t=1.$

███████ ██████████ ███████████████ are █████ in the ███████ ██████████ █████████ below.

█████████ █████████

For the █████████ ${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1$ , ██████ the ████ on the █████ to ██████ the ████████ of the █████████ █████████:

{\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=-1

or ████████████:

{\frac {y^{2}}{b^{2}}}-{\frac {x^{2}}{a^{2}}}=1

A █████████ and ███ █████████ ███ have █████████ which are █████████. In the ██████ of ███████ ██████████, ████ █████████ ███ █████████ ████ of ████ and █████, █████ ███ █████████ ██████████ ██████ at a █████ ███████ ████████ from the ██████, and the █████ ██████████ ██████ at a █████████████ ████████ ████████ from the ██████.

xy=c^{2}

and

xy=-c^{2}

████ ███████ █████████ ██████████.

In █████ ███████████

██████ at the █████

The █████ ███████████ used ████ ████████ for the █████████ are ███████ ████████ to the █████████ ██████████ ██████ that has ███ ██████ in a █████ and ███ █-████ ████████ toward the ██████ of the "█████████ ██████████ ██████" as ███████████ in the █████ ███████.

In this ████ the █████ $\varphi$ is ██████ ████ ███████.

████████ to this ██████████ ██████ ███ has that

$r={\frac {p}{1\mp e\cos \varphi }},\quad p={\frac {b^{2}}{a}}$

and

$-\arccos \left(-{\frac {1}{e}}\right)<\varphi <\arccos \left(-{\frac {1}{e}}\right).$

██████ at the ██████

With █████ ███████████ ████████ to the "█████████ ██████████ ██████" (███ ██████ ███████) ███ has that

$r={\frac {b}{\sqrt {e^{2}\cos ^{2}\varphi -1}}}.\,$

For the █████ ██████ of the █████████ the █████ of $\varphi$ is $-\arccos \left({\frac {1}{e}}\right)<\varphi <\arccos \left({\frac {1}{e}}\right).$

████████████

████ using █████ ███████████, the ████████████ of the █████████ can be █████████ as $\sec \varphi _{\text{max}}$ █████ $\varphi _{\text{max}}$ is the █████ of the ███████ ██████████. As $\varphi$ ██████████ this █████, █ ██████████ ████████ and the ███████████ in ██████ of the █████████ █████ above ██████████ ████, █████:

$e^{2}\cos ^{2}\varphi _{\text{max}}-1=0$

$1\pm e\cos \varphi _{\text{max}}=0$

$\implies e=\sec \varphi _{\text{max}}$

██████████ █████████

A █████████ with ████████ ${\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1$ can be █████████ by ███████ ██████████ █████████:

Through ██████████ █████████████ █████████ ${\begin{cases}x=\pm a\cosh t,\\y=b\sinh t,\end{cases}}\qquad t\in \mathbb {R} .$
As a ████████ ██████████████ ${\begin{cases}x=\pm a{\dfrac {t^{2}+1}{2t}},\\[1ex]y=b{\dfrac {t^{2}-1}{2t}},\end{cases}}\qquad t>0$
Through ████████ █████████████ █████████ ${\begin{cases}x={\frac {a}{\cos t}}=a\sec t,\\y=\pm b\tan t,\end{cases}}\qquad 0\leq t<2\pi ,\ t\neq {\frac {\pi }{2}},\ t\neq {\frac {3}{2}}\pi .$
With the ███████ █████ as █████████: A ██████████ ██████████████, which uses the █████ $█$ of the ███████ at a █████ of the █████████ can be ████████ ███████████ to the ███████ ████: ███████ in the ███████ ████ $b^{2}$ by $-b^{2}$ and use ████████ for the ██████████ █████████. ███ ████ ${\vec {c}}_{\pm }(m)=\left(-{\frac {ma^{2}}{\pm {\sqrt {m^{2}a^{2}-b^{2}}}}},{\frac {-b^{2}}{\pm {\sqrt {m^{2}a^{2}-b^{2}}}}}\right),\quad |m|>b/a.$ ████, ${\vec {c}}_{-}$ is the █████, and ${\vec {c}}_{+}$ the █████ ████ of the █████████. The ██████ with ████████ ████████ (████████ $(\pm a,0)$ ) are not ███████ by the ██████████████. The ████████ of the ███████ at █████ ${\vec {c}}_{\pm }(m)$ is $y=mx\pm {\sqrt {m^{2}a^{2}-b^{2}}}.$ This ███████████ of the ████████ of a █████████ is an █████████ ████ for the █████████████ of the █████████ of a █████████.

██████████ █████████

████ as the █████████████ █████████ are ███████ in █████ of the ████ ██████, ██ ████ the ██████████ █████████ are ███████ in █████ of the ████ █████████, as █████ in this ███████. In a ████ ██████, the █████ (in ███████) is █████ to █████ the ████ of the ████████ ██████ which that █████ ████████. The █████████ ██████████ █████ is ████████ ███████ as █████ the ████ of a ██████████ ██████.

███ $a$ be █████ the ████ between the $█$ ████ and a ███ through the ██████ ████████████ the ████ █████████, and ██████ ${\textstyle (x,y)=(\cosh a,\sinh a)=(x,{\sqrt {x^{2}-1}})}$ as the ███████████ of the ████████████ █████. ████ the ████ of the ██████████ ██████ is the ████ of the ████████ minus the ██████ ██████ past the ██████ at $(1,0)$ : ${\begin{aligned}{\frac {a}{2}}&={\frac {xy}{2}}-\int _{1}^{x}{\sqrt {t^{2}-1}}\,dt\\[1ex]&={\frac {1}{2}}\left(x{\sqrt {x^{2}-1}}\right)-{\frac {1}{2}}\left(x{\sqrt {x^{2}-1}}-\ln \left(x+{\sqrt {x^{2}-1}}\right)\right),\end{aligned}}$ which ██████████ to the ████ ██████████ ██████ $a=\operatorname {arcosh} x=\ln \left(x+{\sqrt {x^{2}-1}}\right).$ ███████ for $█$ ██████ the ███████████ ████ of the ██████████ ██████: $x=\cosh a={\frac {e^{a}+e^{-a}}{2}}.$ From $x^{2}-y^{2}=1$ ███ ████ $y=\sinh a={\sqrt {\cosh ^{2}a-1}}={\frac {e^{a}-e^{-a}}{2}},$ and ███ ███████ the ████ ██████████ ████: $a=\operatorname {arsinh} y=\ln \left(y+{\sqrt {y^{2}+1}}\right).$ █████ ██████████ █████████ are ███████ █████████ to the ██████████ ██████ and ██████████ ████, ██ for ███████ $\operatorname {tanh} a={\frac {\sinh a}{\cosh a}}={\frac {e^{2a}-1}{e^{2a}+1}}.$

██████████

██████████ ████████

The ███████ at a █████ $█$ ███████ the █████ between the █████ ${\overline {PF_{1}}},{\overline {PF_{2}}}.$ This is ██████ the ███████ ████████ or ██████████ ████████ of a █████████.

█████

███ $█$ be the █████ on the ████ ${\overline {PF_{2}}}$ with the ████████ $2a$ to the █████ $F_{2}$ (███ ███████, $a$ is the ████ █████ ████ of the █████████). ████ $█$ is the ████████ of the █████ between the █████ ${\overline {PF_{1}}},{\overline {PF_{2}}}$ . In █████ to █████ that $█$ is the ███████ ████ at █████ $█$ , ███ ██████ that ███ █████ $█$ on ████ $█$ which is █████████ from $█$ ██████ be on the █████████. █████ $█$ has ████ █████ $█$ in ██████ with the █████████ and is, █████████, the ███████ at █████ $█$ . From the ███████ and the ████████ ██████████ ███ ██████████ that $|QF_{2}|<|LF_{2}|+|QL|=2a+|QF_{1}|$ █████, which █████: $|QF_{2}|-|QF_{1}|<2a$ . But if $█$ is a █████ of the █████████, the ██████████ ██████ be $2a$ .

█████████ of ████████ ██████

The █████████ of ████████ ██████ of a █████████ ███ on a ████ through the ██████ (███ ███████).

The ██████ of ███ █████ ███ ███ on █████████ ████████ of the █████████.

The █████ of the ████████ on █████████ is ████ ████ for the █████████ $y=1/x$ . Because ███ █████████ is an ██████ █████ of the █████████ $y=1/x$ (███ ███████ below) and an ██████ ██████████████ █████████ ███████████ and █████████ of ████ ████████, the ████████ is ████ for ███ ██████████: For ███ ██████ $P=\left(x_{1},{\tfrac {1}{x_{1}}}\right),\ Q=\left(x_{2},{\tfrac {1}{x_{2}}}\right)$ of the █████████ $y=1/x$

the ████████ of the █████ is

M=\left({\tfrac {x_{1}+x_{2}}{2}},\cdots \right)=\cdots ={\tfrac {x_{1}+x_{2}}{2}}\;\left(1,{\tfrac {1}{x_{1}x_{2}}}\right)\ ;

;}

the █████ of the █████ is

{\frac {{\tfrac {1}{x_{2}}}-{\tfrac {1}{x_{1}}}}{x_{2}-x_{1}}}=\cdots =-{\tfrac {1}{x_{1}x_{2}}}\ .

For ████████ ██████ the █████ is ████████ and the █████████ of the ████████ ██████ ███ on the ████ $y={\tfrac {1}{x_{1}x_{2}}}\;x\ .$

███████████: for ███ ████ of ██████ $█, █$ of a █████ there ██████ a ████ ██████████ with an ████ (███ of █████ ██████) ███████ through the ██████ of the █████████, which █████████ the ██████ $█, █$ and ██████ the █████████ (as a █████) █████. A ████ ██████████ is a ██████████████ of an ████████ ██████████ across a ████ $█$ , █████ ███ █████-█████ █████ are on a ████ █████████████ to $█$ .

Because a ████ ██████████ ██████ the █████████ █████, the ████ of ██████████ is █████, ███. █████ the ████████ $█$ of a █████ $█ █$ ███████ the ███████ ████ ███████ ${\overline {P}}\,{\overline {Q}}$ between the ██████████ into ██████, ███. This █████ that $|P{\overline {P}}|=|Q{\overline {Q}}|$ . This ████████ can be used for the ████████████ of ███████ ██████ $█$ of the █████████ if a █████ $█$ and the ██████████ are █████.

If the █████ ███████████ into a ███████, ████ the ████████ █████ ███████ the ████ ███████ between the ██████████ in ███ ██████.

██████████ ████████ – █████████

For a █████████ ${\textstyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1,\,a>b}$ the ████████████ ██████ of ██████████ ████████ ███ on the ██████ $x^{2}+y^{2}=a^{2}-b^{2}$ . This ██████ is ██████ the █████████ of the █████ █████████.

The ████████ ███ ██████ to ██████ on █████████ ████████ of the █████████.

In ████ of $a\leq b$ there are ██ █████ of ██████████ ████████.

████-█████ ████████ for a █████████

███ █████████ can be █████████ in a ████████ ██████████ ██████ by an ████████ ${\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1$ . The ████████ of the ███████ at a █████ $P_{0}=(x_{0},y_{0})$ of the █████████ is ${\tfrac {x_{0}x}{a^{2}}}-{\tfrac {y_{0}y}{b^{2}}}=1.$ If ███ ██████ █████ $P_{0}=(x_{0},y_{0})$ to be an █████████ █████ █████████ from the ██████, ████

█████

P_{0}=(x_{0},y_{0})\neq (0,0)

is ██████ onto the ████

{\frac {x_{0}x}{a^{2}}}-{\frac {y_{0}y}{b^{2}}}=1

, not through the ██████ of the █████████.

This ████████ between ██████ and █████ is a █████████.

The ███████ ████████ ████

████

y=mx+d,\ d\neq 0

onto the █████

\left(-{\frac {ma^{2}}{d}},-{\frac {b^{2}}{d}}\right)

and

████

x=c,\ c\neq 0

onto the █████

\left({\frac {a^{2}}{c}},0\right)\ .

████ a ████████ between ██████ and █████ █████████ by a █████ is ██████ ████-█████ ████████ or ████ ████████. The ████ is the █████, the █████ the ████. ███ ████ and █████.

By ███████████ ███ ██████ the following ██████████ of the ████-█████ ████████ of the █████████:

For a █████ (████) on the █████████ the █████ is the ███████ at this █████ (███ ███████: $P_{1},\ p_{1}$ ).
For a ████ $█$ outside the █████████ the ████████████ ██████ of ███ █████ with the █████████ are the ████████ ██████ of the ███ ████████ ███████ $█$ (███ ███████: $P_{2},\ p_{2},\ P_{3},\ p_{3}$ ).
For a █████ within the █████████ the █████ has ██ █████ with the █████████ in ██████. (███ ███████: $P_{4},\ p_{4}$ ).

███████:

The ████████████ █████ of ███ ██████ (for ███████: $p_{2},p_{3}$ ) is the ████ of the ████ through █████ █████ (████: $P_{2},P_{3}$ ).
The ████ $(c,0),$ and $(-c,0)$ ████████████ and the ███████████ $x={\tfrac {a^{2}}{c}}$ and $x=-{\tfrac {a^{2}}{c}}$ ████████████ ██████ to █████ of ████ and █████.

████-█████ █████████ █████ for ████████ and █████████, ███.

█████ ██████████

The following are ██████████: (█) a ██████ ███████ through the █████████'s ████ and ████████ at the █████████'s ██████; (█) ██████ of the █████ that are ███████ to the █████████ at the ████████; and (█) ██████ of the ██████████ of the █████████.
The following are ████ ██████████: (█) the ██████ that is ████████ at the █████████'s ██████ and that ██████ through the █████████'s ████████; (█) ██████ █████████; and (█) ██████ of the ██████████.
Since ████ the ██████████ ████ and the █████████ ████ are ████ of ████████, the ████████ █████ of a █████████ is the █████ ████-█████.
The ███████████ ██████████ ██ = ████████ █████ █████ ███████ by ███████ ████████ which have the ██████████ as █████████ ████.
The ██████ of a ███████████ █████████ is the ████████ █████ of ███ ████ ██████ on it that do not ████ an ████████████ ██████.

███ ██████

The ███ ██████ of a █████████ ████ not have an ██████████ ██████████. The █████ ████ of a █████████ can be █████████████ as

$y=b{\sqrt {{\frac {x^{2}}{a^{2}}}-1}}.$

████ the ████████ ██████ the ███ ██████ $█$ from $x_{1}$ to $x_{2}$ can be ████████ as: