 \text{Hyp: }y=±\sqrt{x²+1}

\sqrt{r²+1})

x

a_1

\sqrt{r²+1})

\triangle SOA_1∼\triangle ARA_1

\frac{a_1}{r-a_1}=\frac{1}{\sqrt{r^{2}+1}}

a_1{\sqrt{r^{2}+1}}=r-a_1

a_1=\frac{r}{({\sqrt{r^{2}+1}}+1)}=\frac{({\sqrt{r^{2}+1}}-1)}{r}

A_1'

\a_2

a_2

a_1×a_2=\frac{1}{r}(\sqrt{r²+1}-1)×\frac{1}{r}(\sqrt{r²+1}+1)=1

ω = Circle((0, 0), 1)
O = Intersect(xAxis, yAxis)
Hyp: x^(2) - y^(2) = -1
S = Intersect(Hyp, yAxis, 2)
S1: (0, -1)
R = Point(xAxis)
a: sqrt(1 + (x(R))^(2))
a_0: x(R)
text1: "R(r,0)=(" + (LaTeX[a_0]) + ",0)"
A: (x(R), a)
f = Segment(R, A)
text2: "A(r,\sqrt{r²+1})=(" + (LaTeX[a_0]) + "," + (LaTeX[a]) + ")"
A' = Mirror(A, O)
g = Segment(A', A)
text7: "A'(-r,-\sqrt{r²+1})=(-" + (LaTeX[a_0]) + " ,-" + (LaTeX[a]) + " )"
h = Line(S, A)
A_1 = Intersect(h, xAxis)
j = Segment(O, A_1)
l = Segment(R, A)
C = (2.0899924868519917, -1.175251690458304)
text4: "{A_1}(\frac{1}{r}(\sqrt{r²+1}-1),0)=" + (LaTeX[A_1]) + ""
u = Vector(C, A_1)
i = Line(S, A')
B = Intersect(i, xAxis)
k = Segment(O, B)
D = (4.343936889556725, -2.0016979714500387)
v = Vector(D, B)
text5: "{{A'_1}}(\frac{1}{r}(\sqrt{r²+1}+1),0
)=" + (LaTeX[B]) + " "