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(
u
+
v
)
2
(u+v)^2
(
u
+
v
)
2
(
u
−
v
)
2
(u-v)^2
(
u
−
v
)
2
u
,
v
u,v
u
,
v
(
u
+
(
−
v
)
)
2
(u+(-v))^2
(
u
+
(
−
v
)
)
2
v
v
v
(
u
+
v
)
2
=
u
2
+
v
2
+
2
u
v
(
u
−
v
)
2
=
u
2
+
v
2
−
2
u
v
\begin{aligned} (u+v)^2&=u^2+v^2+2uv\\ (u-v)^2&=u^2+v^2-2uv \end{aligned}
(
u
+
v
)
2
(
u
−
v
)
2
=
u
2
+
v
2
+
2
uv
=
u
2
+
v
2
−
2
uv
u
2
+
v
2
u^2+v^2
u
2
+
v
2
+
2
u
v
+2uv
+
2
uv
−
2
u
v
-2uv
−
2
uv
(
−
v
)
2
=
v
2
(-v)^2=v^2
(
−
v
)
2
=
v
2
2
u
(
−
v
)
=
−
2
u
v
2u(-v)=-2uv
2
u
(
−
v
)
=
−
2
uv
u
2
−
v
2
u^2-v^2
u
2
−
v
2
u
2
+
v
2
+
2
u
v
u^2+v^2+2uv
u
2
+
v
2
+
2
uv
u
2
+
v
2
−
2
u
v
u^2+v^2-2uv
u
2
+
v
2
−
2
uv
2
u
v
+
(
−
2
u
v
)
=
0
2uv+(-2uv)=0
2
uv
+
(
−
2
uv
)
=
0
2
u
v
−
(
−
2
u
v
)
=
4
u
v
2uv-(-2uv)=4uv
2
uv
−
(
−
2
uv
)
=
4
uv
(
u
+
v
)
2
+
(
u
−
v
)
2
=
2
(
u
2
+
v
2
)
(
u
+
v
)
2
−
(
u
−
v
)
2
=
4
u
v
\begin{aligned} (u+v)^2+(u-v)^2&=2(u^2+v^2)\\ (u+v)^2-(u-v)^2&=4uv \end{aligned}
(
u
+
v
)
2
+
(
u
−
v
)
2
(
u
+
v
)
2
−
(
u
−
v
)
2
=
2
(
u
2
+
v
2
)
=
4
uv
2
u
v
2uv
2
uv
2
(
u
2
+
v
2
)
2(u^2+v^2)
2
(
u
2
+
v
2
)
4
u
v
4uv
4
uv
(
u
+
v
)
2
+
(
u
−
v
)
2
=
2
(
u
2
+
v
2
)
(u+v)^2+(u-v)^2=2(u^2+v^2)
(
u
+
v
)
2
+
(
u
−
v
)
2
=
2
(
u
2
+
v
2
)
(
u
+
v
)
2
−
(
u
−
v
)
2
=
4
u
v
(u+v)^2-(u-v)^2=4uv
(
u
+
v
)
2
−
(
u
−
v
)
2
=
4
uv
v
↦
−
v
v\mapsto-v
v
↦
−
v
Q
+
=
(
u
+
v
)
2
Q_+=(u+v)^2
Q
+
=
(
u
+
v
)
2
Q
−
=
(
u
−
v
)
2
Q_-=(u-v)^2
Q
−
=
(
u
−
v
)
2
−
v
-v
−
v
Q
+
+
Q
−
2
\frac{Q_++Q_-}{2}
2
Q
+
+
Q
−
Q
+
−
Q
−
2
\frac{Q_+-Q_-}{2}
2
Q
+
−
Q
−
Q
+
Q_+
Q
+
Q
−
Q_-
Q
−
v
↦
−
v
v\mapsto -v
v
↦
−
v
A
B
C
D
ABCD
A
BC
D
A
(
0
,
0
)
A(0,0)
A
(
0
,
0
)
B
(
a
,
0
)
B(a,0)
B
(
a
,
0
)
D
(
b
,
c
)
D(b,c)
D
(
b
,
c
)
a
,
b
,
c
a,b,c
a
,
b
,
c
C
C
C
∣
A
C
∣
2
|AC|^2
∣
A
C
∣
2
∣
B
D
∣
2
|BD|^2
∣
B
D
∣
2
∣
A
B
∣
2
|AB|^2
∣
A
B
∣
2
∣
A
D
∣
2
|AD|^2
∣
A
D
∣
2
D
C
→
=
A
B
→
\overrightarrow{DC}=\overrightarrow{AB}
D
C
=
A
B
C
(
a
+
b
,
c
)
C(a+b,c)
C
(
a
+
b
,
c
)
∣
A
C
∣
2
=
(
a
+
b
)
2
+
c
2
∣
B
D
∣
2
=
(
b
−
a
)
2
+
c
2
∣
A
B
∣
2
=
a
2
∣
A
D
∣
2
=
b
2
+
c
2
\begin{aligned} |AC|^2&=(a+b)^2+c^2\\ |BD|^2&=(b-a)^2+c^2\\ |AB|^2&=a^2\\ |AD|^2&=b^2+c^2 \end{aligned}
∣
A
C
∣
2
∣
B
D
∣
2
∣
A
B
∣
2
∣
A
D
∣
2
=
(
a
+
b
)
2
+
c
2
=
(
b
−
a
)
2
+
c
2
=
a
2
=
b
2
+
c
2
(
b
−
a
)
2
=
(
a
−
b
)
2
(b-a)^2=(a-b)^2
(
b
−
a
)
2
=
(
a
−
b
)
2
(
a
+
b
)
2
(a+b)^2
(
a
+
b
)
2
(
a
−
b
)
2
(a-b)^2
(
a
−
b
)
2
c
2
c^2
c
2
(
a
−
b
,
c
)
(a-b,c)
(
a
−
b
,
c
)
(
a
+
b
,
0
)
(a+b,0)
(
a
+
b
,
0
)
(
a
+
b
,
c
)
(a+b,c)
(
a
+
b
,
c
)
(
a
+
b
)
2
+
c
2
(a+b)^2+c^2
(
a
+
b
)
2
+
c
2
(
a
−
b
)
2
+
c
2
(a-b)^2+c^2
(
a
−
b
)
2
+
c
2
a
2
+
b
2
+
c
2
a^2+b^2+c^2
a
2
+
b
2
+
c
2
b
2
+
c
2
b^2+c^2
b
2
+
c
2
(
b
−
a
)
2
+
c
2
(b-a)^2+c^2
(
b
−
a
)
2
+
c
2
a
a
a
a
2
+
c
2
a^2+c^2
a
2
+
c
2
a
2
a^2
a
2
(
b
+
c
)
2
(b+c)^2
(
b
+
c
)
2
∣
A
C
∣
2
=
(
a
+
b
)
2
+
c
2
|AC|^2=(a+b)^2+c^2
∣
A
C
∣
2
=
(
a
+
b
)
2
+
c
2
∣
B
D
∣
2
=
(
b
−
a
)
2
+
c
2
|BD|^2=(b-a)^2+c^2
∣
B
D
∣
2
=
(
b
−
a
)
2
+
c
2
∣
A
B
∣
2
=
a
2
|AB|^2=a^2
∣
A
B
∣
2
=
a
2
∣
A
D
∣
2
=
b
2
+
c
2
|AD|^2=b^2+c^2
∣
A
D
∣
2
=
b
2
+
c
2
∣
A
C
∣
2
+
∣
B
D
∣
2
=
2
(
∣
A
B
∣
2
+
∣
A
D
∣
2
)
|AC|^2+|BD|^2=2(|AB|^2+|AD|^2)
∣
A
C
∣
2
+
∣
B
D
∣
2
=
2
(
∣
A
B
∣
2
+
∣
A
D
∣
2
)
(
b
−
a
)
2
(b-a)^2
(
b
−
a
)
2
∣
A
C
∣
2
+
∣
B
D
∣
2
=
(
a
+
b
)
2
+
(
a
−
b
)
2
+
2
c
2
=
2
(
a
2
+
b
2
)
+
2
c
2
=
2
(
a
2
+
(
b
2
+
c
2
)
)
=
2
(
∣
A
B
∣
2
+
∣
A
D
∣
2
)
\begin{aligned} |AC|^2+|BD|^2&=(a+b)^2+(a-b)^2+2c^2\\ &=2(a^2+b^2)+2c^2\\ &=2\bigl(a^2+(b^2+c^2)\bigr)\\ &=2(|AB|^2+|AD|^2) \end{aligned}
∣
A
C
∣
2
+
∣
B
D
∣
2
=
(
a
+
b
)
2
+
(
a
−
b
)
2
+
2
c
2
=
2
(
a
2
+
b
2
)
+
2
c
2
=
2
(
a
2
+
(
b
2
+
c
2
)
)
=
2
(
∣
A
B
∣
2
+
∣
A
D
∣
2
)
u
=
a
u=a
u
=
a
v
=
b
v=b
v
=
b
2
c
2
2c^2
2
c
2
(
a
,
b
)
(a,b)
(
a
,
b
)
∣
A
C
∣
2
+
∣
B
D
∣
2
=
2
(
∣
A
B
∣
2
+
∣
A
D
∣
2
)
|AC|^2+|BD|^2=2(|AB|^2+|AD|^2)
∣
A
C
∣
2
+
∣
B
D
∣
2
=
2
(
∣
A
B
∣
2
+
∣
A
D
∣
2
)
∣
A
C
∣
2
−
∣
B
D
∣
2
|AC|^2-|BD|^2
∣
A
C
∣
2
−
∣
B
D
∣
2
4
A
B
→
⋅
A
D
→
4\overrightarrow{AB}\cdot\overrightarrow{AD}
4
A
B
⋅
A
D
a
b
<
0
ab<0
ab
<
0
a
b
+
0
c
a b+0 c
ab
+
0
c
∣
A
C
∣
2
−
∣
B
D
∣
2
=
(
a
+
b
)
2
−
(
a
−
b
)
2
=
4
a
b
\begin{aligned} |AC|^2-|BD|^2&=(a+b)^2-(a-b)^2\\ &=4ab \end{aligned}
∣
A
C
∣
2
−
∣
B
D
∣
2
=
(
a
+
b
)
2
−
(
a
−
b
)
2
=
4
ab
A
B
→
=
(
a
,
0
)
\overrightarrow{AB}=(a,0)
A
B
=
(
a
,
0
)
A
D
→
=
(
b
,
c
)
\overrightarrow{AD}=(b,c)
A
D
=
(
b
,
c
)
A
B
→
⋅
A
D
→
=
a
b
\overrightarrow{AB}\cdot\overrightarrow{AD}=ab
A
B
⋅
A
D
=
ab
+
4
+4
+
4
−
4
-4
−
4
∣
A
C
∣
<
∣
B
D
∣
|AC|<|BD|
∣
A
C
∣
<
∣
B
D
∣
∣
A
C
∣
2
−
∣
B
D
∣
2
=
4
A
B
→
⋅
A
D
→
|AC|^2-|BD|^2=4\overrightarrow{AB}\cdot\overrightarrow{AD}
∣
A
C
∣
2
−
∣
B
D
∣
2
=
4
A
B
⋅
A
D
±
\pm
±
∣
B
D
∣
2
−
∣
A
C
∣
2
=
0
|BD|^2-|AC|^2=0
∣
B
D
∣
2
−
∣
A
C
∣
2
=
0
∣
B
D
∣
2
−
∣
A
C
∣
2
=
−
4
A
B
→
⋅
A
D
→
|BD|^2-|AC|^2=-4\overrightarrow{AB}\cdot\overrightarrow{AD}
∣
B
D
∣
2
−
∣
A
C
∣
2
=
−
4
A
B
⋅
A
D
∣
B
D
∣
2
−
∣
A
C
∣
2
=
4
A
B
→
⋅
A
D
→
|BD|^2-|AC|^2=4\overrightarrow{AB}\cdot\overrightarrow{AD}
∣
B
D
∣
2
−
∣
A
C
∣
2
=
4
A
B
⋅
A
D
4
a
b
4ab
4
ab
x
1
<
x
2
x_1<x_2
x
1
<
x
2
x
2
−
8
x
+
12
=
0
x^2-8x+12=0
x
2
−
8
x
+
12
=
0
x
1
+
x
2
=
8
x_1+x_2=8
x
1
+
x
2
=
8
x
1
x
2
=
12
x_1x_2=12
x
1
x
2
=
12
x
1
=
4
−
d
x_1=4-d
x
1
=
4
−
d
x
2
=
4
+
d
x_2=4+d
x
2
=
4
+
d
d
=
x
2
−
x
1
2
>
0
d=\frac{x_2-x_1}{2}>0
d
=
2
x
2
−
x
1
>
0
(
4
−
d
)
(
4
+
d
)
=
16
−
d
2
=
12
⇒
d
=
2
(4-d)(4+d)=16-d^2=12\Rightarrow d=2
(
4
−
d
)
(
4
+
d
)
=
16
−
d
2
=
12
⇒
d
=
2
x
1
=
2
x_1=2
x
1
=
2
x
2
=
6
x_2=6
x
2
=
6
d
>
0
d>0
d
>
0
x
1
x_1
x
1
4
+
d
4+d
4
+
d
8
−
d
8-d
8
−
d
4
−
d
4-d
4
−
d
x
2
x_2
x
2
8
+
d
8+d
8
+
d
d
d
d
−
2
-2
−
2
2
2
2
4
4
4
d
2
=
4
d^2=4
d
2
=
4
d
=
2
d=2
d
=
2
F
1
(
−
c
,
0
)
F_1(-c,0)
F
1
(
−
c
,
0
)
F
2
(
c
,
0
)
F_2(c,0)
F
2
(
c
,
0
)
a
>
c
>
0
a>c>0
a
>
c
>
0
r
1
=
∣
P
F
1
∣
r_1=|PF_1|
r
1
=
∣
P
F
1
∣
r
2
=
∣
P
F
2
∣
r_2=|PF_2|
r
2
=
∣
P
F
2
∣
P
(
x
,
y
)
P(x,y)
P
(
x
,
y
)
2
a
2a
2
a
2
c
2c
2
c
r
1
r_1
r
1
P
P
P
−
c
-c
−
c
r
1
+
r
2
=
2
c
r_1+r_2=2c
r
1
+
r
2
=
2
c
r
1
2
=
(
x
+
c
)
2
+
y
2
r_1^2=(x+c)^2+y^2
r
1
2
=
(
x
+
c
)
2
+
y
2
r
2
2
=
(
x
−
c
)
2
+
y
2
r_2^2=(x-c)^2+y^2
r
2
2
=
(
x
−
c
)
2
+
y
2
r
1
+
r
2
=
2
a
r_1+r_2=2a
r
1
+
r
2
=
2
a
r
1
r
2
r_1r_2
r
1
r
2
r
1
2
=
(
x
−
c
)
2
+
y
2
r_1^2=(x-c)^2+y^2
r
1
2
=
(
x
−
c
)
2
+
y
2
r
2
2
=
(
x
+
c
)
2
+
y
2
r_2^2=(x+c)^2+y^2
r
2
2
=
(
x
+
c
)
2
+
y
2
r
1
2
+
r
2
2
+
2
r
1
r
2
=
4
a
2
r_1^2+r_2^2+2r_1r_2=4a^2
r
1
2
+
r
2
2
+
2
r
1
r
2
=
4
a
2
[
(
x
+
c
)
2
+
y
2
]
[
(
x
−
c
)
2
+
y
2
]
\sqrt{[(x+c)^2+y^2][(x-c)^2+y^2]}
[(
x
+
c
)
2
+
y
2
]
[(
x
−
c
)
2
+
y
2
]
r
1
2
,
r
2
2
r_1^2,r_2^2
r
1
2
,
r
2
2
a
+
d
a+d
a
+
d
a
−
d
a-d
a
−
d
d
=
r
1
−
r
2
2
d=\frac{r_1-r_2}{2}
d
=
2
r
1
−
r
2
r
1
,
r
2
r_1,r_2
r
1
,
r
2
a
,
d
a,d
a
,
d
2
r
1
,
2
r
2
2r_1,2r_2
2
r
1
,
2
r
2
r
1
>
r
2
r_1>r_2
r
1
>
r
2
r
1
=
r
2
r_1=r_2
r
1
=
r
2
r
1
<
r
2
r_1<r_2
r
1
<
r
2
r
1
=
a
+
d
r_1=a+d
r
1
=
a
+
d
r
2
=
a
−
d
r_2=a-d
r
2
=
a
−
d
−
d
-d
−
d
∣
r
1
−
r
2
∣
/
2
|r_1-r_2|/2
∣
r
1
−
r
2
∣/2
2
a
+
d
2a+d
2
a
+
d
r
2
r_2
r
2
2
a
−
d
2a-d
2
a
−
d
d
=
(
r
1
−
r
2
)
/
2
d=(r_1-r_2)/2
d
=
(
r
1
−
r
2
)
/2
(
a
+
d
)
2
=
x
2
+
2
c
x
+
c
2
+
y
2
(
a
−
d
)
2
=
x
2
−
2
c
x
+
c
2
+
y
2
\begin{aligned} (a+d)^2&=x^2+2cx+c^2+y^2\\ (a-d)^2&=x^2-2cx+c^2+y^2 \end{aligned}
(
a
+
d
)
2
(
a
−
d
)
2
=
x
2
+
2
c
x
+
c
2
+
y
2
=
x
2
−
2
c
x
+
c
2
+
y
2
x
x
x
x
2
+
c
2
+
y
2
x^2+c^2+y^2
x
2
+
c
2
+
y
2
y
=
0
y=0
y
=
0
(
a
+
d
)
2
=
(
x
+
c
)
2
+
y
2
(
a
−
d
)
2
=
(
x
−
c
)
2
+
y
2
\begin{aligned} (a+d)^2&=(x+c)^2+y^2\\ (a-d)^2&=(x-c)^2+y^2 \end{aligned}
(
a
+
d
)
2
(
a
−
d
)
2
=
(
x
+
c
)
2
+
y
2
=
(
x
−
c
)
2
+
y
2
F
1
(
−
c
,
0
)
,
F
2
(
c
,
0
)
F_1(-c,0),F_2(c,0)
F
1
(
−
c
,
0
)
,
F
2
(
c
,
0
)
r
1
−
r
2
r_1-r_2
r
1
−
r
2
y
2
y^2
y
2
(
u
,
v
)
(u,v)
(
u
,
v
)
(
a
,
d
)
(a,d)
(
a
,
d
)
(
x
,
c
)
(x,c)
(
x
,
c
)
(
a
+
d
)
2
−
(
a
−
d
)
2
=
(
x
+
c
)
2
−
(
x
−
c
)
2
(a+d)^2-(a-d)^2=(x+c)^2-(x-c)^2
(
a
+
d
)
2
−
(
a
−
d
)
2
=
(
x
+
c
)
2
−
(
x
−
c
)
2
4
a
d
=
4
c
x
⇒
a
d
=
c
x
4ad=4cx\Rightarrow ad=cx
4
a
d
=
4
c
x
⇒
a
d
=
c
x
a
>
0
a>0
a
>
0
d
=
c
a
x
d=\frac ca x
d
=
a
c
x
r
1
−
r
2
=
2
d
=
2
c
a
x
r_1-r_2=2d=\frac{2c}{a}x
r
1
−
r
2
=
2
d
=
a
2
c
x
4
a
d
=
4
c
x
4ad=4cx
4
a
d
=
4
c
x
d
=
(
c
/
a
)
x
d=(c/a)x
d
=
(
c
/
a
)
x
x
=
0
x=0
x
=
0
y
y
y
c
a
\frac ca
a
c
1
1
1
−
x
-x
−
x
d
=
x
d=x
d
=
x
r
1
−
r
2
=
2
c
r_1-r_2=2c
r
1
−
r
2
=
2
c
c
/
a
>
0
c/a>0
c
/
a
>
0
r
1
−
r
2
=
(
2
c
/
a
)
x
r_1-r_2=(2c/a)x
r
1
−
r
2
=
(
2
c
/
a
)
x
r
1
2
+
r
2
2
r_1^2+r_2^2
r
1
2
+
r
2
2
u
=
r
1
u=r_1
u
=
r
1
v
=
r
2
v=r_2
v
=
r
2
4
r
1
r
2
4r_1r_2
4
r
1
r
2
(
r
1
+
r
2
)
2
+
(
r
1
−
r
2
)
2
=
r
1
2
+
r
2
2
(r_1+r_2)^2+(r_1-r_2)^2=r_1^2+r_2^2
(
r
1
+
r
2
)
2
+
(
r
1
−
r
2
)
2
=
r
1
2
+
r
2
2
r
1
+
r
2
=
r
1
2
+
r
2
2
r_1+r_2=r_1^2+r_2^2
r
1
+
r
2
=
r
1
2
+
r
2
2
(
r
1
+
r
2
)
2
+
(
r
1
−
r
2
)
2
=
2
(
r
1
2
+
r
2
2
)
(r_1+r_2)^2+(r_1-r_2)^2=2(r_1^2+r_2^2)
(
r
1
+
r
2
)
2
+
(
r
1
−
r
2
)
2
=
2
(
r
1
2
+
r
2
2
)
(
r
1
+
r
2
)
2
−
(
r
1
−
r
2
)
2
=
2
(
r
1
2
+
r
2
2
)
(r_1+r_2)^2-(r_1-r_2)^2=2(r_1^2+r_2^2)
(
r
1
+
r
2
)
2
−
(
r
1
−
r
2
)
2
=
2
(
r
1
2
+
r
2
2
)
(
r
1
,
r
2
)
(r_1,r_2)
(
r
1
,
r
2
)
r
1
−
r
2
=
2
c
a
x
r_1-r_2=\frac{2c}{a}x
r
1
−
r
2
=
a
2
c
x
L
=
(
r
1
+
r
2
)
2
+
(
r
1
−
r
2
)
2
R
=
2
(
r
1
2
+
r
2
2
)
\begin{aligned} L&=(r_1+r_2)^2+(r_1-r_2)^2\\ R&=2(r_1^2+r_2^2) \end{aligned}
L
R
=
(
r
1
+
r
2
)
2
+
(
r
1
−
r
2
)
2
=
2
(
r
1
2
+
r
2
2
)
L
L
L
R
R
R
L
=
R
L=R
L
=
R
(
x
+
c
)
2
+
(
x
−
c
)
2
(x+c)^2+(x-c)^2
(
x
+
c
)
2
+
(
x
−
c
)
2
L
=
(
2
a
)
2
+
(
2
c
a
x
)
2
L=(2a)^2+\left(\frac{2c}{a}x\right)^2
L
=
(
2
a
)
2
+
(
a
2
c
x
)
2
r
1
2
+
r
2
2
=
(
x
+
c
)
2
+
(
x
−
c
)
2
+
2
y
2
=
2
(
x
2
+
c
2
)
+
2
y
2
=
2
(
x
2
+
y
2
+
c
2
)
\begin{aligned} r_1^2+r_2^2&=(x+c)^2+(x-c)^2+2y^2\\ &=2(x^2+c^2)+2y^2\\ &=2(x^2+y^2+c^2) \end{aligned}
r
1
2
+
r
2
2
=
(
x
+
c
)
2
+
(
x
−
c
)
2
+
2
y
2
=
2
(
x
2
+
c
2
)
+
2
y
2
=
2
(
x
2
+
y
2
+
c
2
)
(
2
a
)
2
+
(
2
c
a
x
)
2
=
4
(
x
2
+
y
2
+
c
2
)
(2a)^2+\left(\frac{2c}{a}x\right)^2=4(x^2+y^2+c^2)
(
2
a
)
2
+
(
a
2
c
x
)
2
=
4
(
x
2
+
y
2
+
c
2
)
a
2
+
c
2
a
2
x
2
=
x
2
+
y
2
+
c
2
a^2+\frac{c^2}{a^2}x^2=x^2+y^2+c^2
a
2
+
a
2
c
2
x
2
=
x
2
+
y
2
+
c
2
2
y
2
2y^2
2
y
2
a
2
+
(
c
2
/
a
2
)
x
2
=
x
2
+
y
2
+
c
2
a^2+(c^2/a^2)x^2=x^2+y^2+c^2
a
2
+
(
c
2
/
a
2
)
x
2
=
x
2
+
y
2
+
c
2
b
b
b
b
>
0
b>0
b
>
0
a
2
+
c
2
a
2
x
2
=
x
2
+
y
2
+
c
2
a^2+\frac{c^2}{a^2}x^2=x^2+y^2+c^2
a
2
+
a
2
c
2
x
2
=
x
2
+
y
2
+
c
2
a
,
c
a,c
a
,
c
x
2
x^2
x
2
a
2
−
c
2
a^2-c^2
a
2
−
c
2
b
2
b^2
b
2
(
1
−
c
2
a
2
)
x
2
+
y
2
=
a
2
−
c
2
\left(1-\frac{c^2}{a^2}\right)x^2+y^2=a^2-c^2
(
1
−
a
2
c
2
)
x
2
+
y
2
=
a
2
−
c
2
x
2
a
2
+
y
2
a
2
−
c
2
=
1
\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1
a
2
x
2
+
a
2
−
c
2
y
2
=
1
a
2
−
c
2
>
0
a^2-c^2>0
a
2
−
c
2
>
0
b
=
a
2
−
c
2
>
0
b=\sqrt{a^2-c^2}>0
b
=
a
2
−
c
2
>
0
x
2
a
2
+
y
2
b
2
=
1
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1
a
2
x
2
+
b
2
y
2
=
1
a
>
b
>
0
a>b>0
a
>
b
>
0
c
>
0
c>0
c
>
0
b
<
a
b<a
b
<
a
b
2
=
a
2
−
c
2
b^2=a^2-c^2
b
2
=
a
2
−
c
2
x
2
a
2
+
y
2
b
2
=
1
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1
a
2
x
2
+
b
2
y
2
=
1
s
2
=
t
2
s^2=t^2
s
2
=
t
2
s
=
t
s=t
s
=
t
∣
x
∣
≤
a
|x|\le a
∣
x
∣
≤
a
a
±
c
a
x
≥
a
−
c
>
0
a\pm\frac ca x\ge a-c>0
a
±
a
c
x
≥
a
−
c
>
0
r
1
=
a
+
(
c
/
a
)
x
r_1=a+(c/a)x
r
1
=
a
+
(
c
/
a
)
x
r
2
=
a
−
(
c
/
a
)
x
r_2=a-(c/a)x
r
2
=
a
−
(
c
/
a
)
x
(
a
±
(
c
/
a
)
x
)
2
(a\pm(c/a)x)^2
(
a
±
(
c
/
a
)
x
)
2
a
±
(
c
/
a
)
x
>
0
a\pm(c/a)x>0
a
±
(
c
/
a
)
x
>
0
y
2
=
(
a
2
−
c
2
)
(
1
−
x
2
a
2
)
y^2=(a^2-c^2)\left(1-\frac{x^2}{a^2}\right)
y
2
=
(
a
2
−
c
2
)
(
1
−
a
2
x
2
)
(
x
+
c
)
2
+
y
2
=
x
2
+
2
c
x
+
c
2
+
(
a
2
−
c
2
)
(
1
−
x
2
a
2
)
=
a
2
+
2
c
x
+
c
2
a
2
x
2
=
(
a
+
c
a
x
)
2
\begin{aligned} (x+c)^2+y^2&=x^2+2cx+c^2+(a^2-c^2)\left(1-\frac{x^2}{a^2}\right)\\ &=a^2+2cx+\frac{c^2}{a^2}x^2\\ &=\left(a+\frac ca x\right)^2 \end{aligned}
(
x
+
c
)
2
+
y
2
=
x
2
+
2
c
x
+
c
2
+
(
a
2
−
c
2
)
(
1
−
a
2
x
2
)
=
a
2
+
2
c
x
+
a
2
c
2
x
2
=
(
a
+
a
c
x
)
2
(
x
−
c
)
2
+
y
2
=
(
a
−
c
a
x
)
2
(x-c)^2+y^2=\left(a-\frac ca x\right)^2
(
x
−
c
)
2
+
y
2
=
(
a
−
a
c
x
)
2
a
±
c
a
x
≥
a
−
c
a
∣
x
∣
≥
a
−
c
>
0
a\pm\frac ca x\ge a-\frac ca|x|\ge a-c>0
a
±
a
c
x
≥
a
−
a
c
∣
x
∣
≥
a
−
c
>
0
∣
P
F
1
∣
=
a
+
c
a
x
|PF_1|=a+\frac ca x
∣
P
F
1
∣
=
a
+
a
c
x
∣
P
F
2
∣
=
a
−
c
a
x
|PF_2|=a-\frac ca x
∣
P
F
2
∣
=
a
−
a
c
x
∣
P
F
1
∣
+
∣
P
F
2
∣
=
2
a
|PF_1|+|PF_2|=2a
∣
P
F
1
∣
+
∣
P
F
2
∣
=
2
a
a
±
(
c
/
a
)
x
≥
a
−
c
>
0
a\pm(c/a)x\ge a-c>0
a
±
(
c
/
a
)
x
≥
a
−
c
>
0
2
(
∥
U
∥
2
+
∥
V
∥
2
)
2(\|\mathbf U\|^2+\|\mathbf V\|^2)
2
(
∥
U
∥
2
+
∥
V
∥
2
)
A
C
AC
A
C
B
D
BD
B
D
4
U
⋅
V
4\mathbf U\cdot\mathbf V
4
U
⋅
V
±
2
r
1
r
2
\pm2r_1r_2
±
2
r
1
r
2
2
(
r
1
2
+
r
2
2
)
2(r_1^2+r_2^2)
2
(
r
1
2
+
r
2
2
)
±
2
c
x
\pm2cx
±
2
c
x
r
1
2
+
r
2
2
=
2
(
x
2
+
y
2
+
c
2
)
r_1^2+r_2^2=2(x^2+y^2+c^2)
r
1
2
+
r
2
2
=
2
(
x
2
+
y
2
+
c
2
)
U
=
A
B
→
\mathbf U=\overrightarrow{AB}
U
=
A
B
V
=
A
D
→
\mathbf V=\overrightarrow{AD}
V
=
A
D
U
+
V
\mathbf U+\mathbf V
U
+
V
V
−
U
\mathbf V-\mathbf U
V
−
U
r
1
+
r
2
,
r
1
−
r
2
r_1+r_2,\ r_1-r_2
r
1
+
r
2
,
r
1
−
r
2
(
x
+
c
)
2
,
(
x
−
c
)
2
(x+c)^2,(x-c)^2
(
x
+
c
)
2
,
(
x
−
c
)
2
a
2
=
b
2
+
c
2
a^2=b^2+c^2
a
2
=
b
2
+
c
2
r
1
+
r
2
r_1+r_2
r
1
+
r
2
F
1
P
→
+
F
2
P
→
\overrightarrow{F_1P}+\overrightarrow{F_2P}
F
1
P
+
F
2
P
P
(
0
,
b
)
P(0,b)
P
(
0
,
b
)
r
1
=
r
2
=
c
2
+
b
2
=
a
r_1=r_2=\sqrt{c^2+b^2}=a
r
1
=
r
2
=
c
2
+
b
2
=
a
F
1
P
→
=
(
c
,
b
)
\overrightarrow{F_1P}=(c,b)
F
1
P
=
(
c
,
b
)
F
2
P
→
=
(
−
c
,
b
)
\overrightarrow{F_2P}=(-c,b)
F
2
P
=
(
−
c
,
b
)
∥
F
1
P
→
+
F
2
P
→
∥
=
2
b
<
2
a
\left\|\overrightarrow{F_1P}+\overrightarrow{F_2P}\right\|=2b<2a
F
1
P
+
F
2
P
=
2
b
<
2
a
c
c
c
2
b
2b
2
b
2
b
<
2
a
2b<2a
2
b
<
2
a
(
a
+
b
,
a
−
b
)
(a+b,a-b)
(
a
+
b
,
a
−
b
)
(
a
+
d
,
a
−
d
)
(a+d,a-d)
(
a
+
d
,
a
−
d
)
(
x
+
c
,
x
−
c
)
(x+c,x-c)
(
x
+
c
,
x
−
c
)
(
r
1
+
r
2
,
r
1
−
r
2
)
(r_1+r_2,r_1-r_2)
(
r
1
+
r
2
,
r
1
−
r
2
)
(
x
+
c
)
2
+
y
2
(x+c)^2+y^2
(
x
+
c
)
2
+
y
2
(
x
−
c
)
2
+
y
2
(x-c)^2+y^2
(
x
−
c
)
2
+
y
2
a
+
d
,
a
−
d
a+d,a-d
a
+
d
,
a
−
d
a
d
=
c
x
ad=cx
a
d
=
c
x
∣
A
C
∣
2
−
∣
B
D
∣
2
=
4
A
B
→
⋅
A
D
→
|AC|^2-|BD|^2=4\overrightarrow{AB}\cdot\overrightarrow{AD}
∣
A
C
∣
2
−
∣
B
D
∣
2
=
4
A
B
⋅
A
D