E
q
u
a
t
i
o
n
Equation
E
q
u
a
t
i
o
n
G
r
a
p
h
Graph
G
r
a
p
h
F
o
c
u
s
Focus
F
o
c
u
s
L
e
n
g
t
h
o
f
L
R
Length\space of\space LR
L
e
n
g
t
h
o
f
L
R
D
i
r
e
c
t
r
i
x
Directrix
D
i
r
e
c
t
r
i
x
L
e
n
g
t
h
o
f
M
a
j
o
r
A
x
i
s
Length\space of\space Major\space Axis
L
e
n
g
t
h
o
f
M
a
j
o
r
A
x
i
s
x
2
a
2
+
y
2
b
2
=
1
\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1
a
2
x
2
+
b
2
y
2
=
1
a
>
b
a>b
a
>
b
(
±
a
e
,
0
)
(±ae,0)
(
±
a
e
,
0
)
2
b
2
a
\frac{2b^2}{a}
a
2
b
2
x
=
±
a
e
x=±\frac{a}{e}
x
=
±
e
a
2
a
2a
2
a
x
2
a
2
+
y
2
b
2
=
1
\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1
a
2
x
2
+
b
2
y
2
=
1
a
<
b
a < b
a
<
b
(
0
,
±
b
e
)
(0,±be)
(
0
,
±
b
e
)
2
a
2
b
\frac{2a^2}{b}
b
2
a
2
y
=
±
b
e
y=±\frac{b}{e}
y
=
±
e
b
2
b
2b
2
b
Equations of Tangent of Ellipse
E
q
u
a
t
i
o
n
Equation
E
q
u
a
t
i
o
n
P
a
r
a
m
e
t
r
i
c
C
o
o
r
d
i
n
a
t
e
s
Parametric\space Coordinates
P
a
r
a
m
e
t
r
i
c
C
o
o
r
d
i
n
a
t
e
s
E
q
u
a
t
i
o
n
o
f
t
a
n
g
e
n
t
Equation\space of\space tangent
E
q
u
a
t
i
o
n
o
f
t
a
n
g
e
n
t
C
o
n
d
i
t
i
o
n
o
f
T
a
n
g
e
n
c
y
Condition\space of \space Tangency
C
o
n
d
i
t
i
o
n
o
f
T
a
n
g
e
n
c
y
x
2
a
2
+
y
2
b
2
=
1
\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1
a
2
x
2
+
b
2
y
2
=
1
a
>
b
a>b
a
>
b
(
a
c
o
s
θ
,
b
s
i
n
θ
)
(acos\theta,bsin\theta)
(
a
c
o
s
θ
,
b
s
i
n
θ
)
y
=
m
x
±
a
m
2
+
b
2
y=mx±\sqrt{am^2+b^2}
y
=
m
x
±
a
m
2
+
b
2
x
c
o
s
θ
a
+
y
s
i
n
θ
b
=
1
\frac{xcos\theta}{a}+\frac{ysin\theta}{b}=1
a
x
c
o
s
θ
+
b
y
s
i
n
θ
=
1
c
=
±
a
m
2
+
b
2
c=±\sqrt{am^2+b^2}
c
=
±
a
m
2
+
b
2
x
2
a
2
+
y
2
b
2
=
1
\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1
a
2
x
2
+
b
2
y
2
=
1
a
<
b
a < b
a
<
b
(
b
c
o
s
θ
,
a
s
i
n
θ
)
(bcos\theta,asin\theta)
(
b
c
o
s
θ
,
a
s
i
n
θ
)
y
=
m
x
±
b
m
2
+
a
2
y=mx±\sqrt{bm^2+a^2}
y
=
m
x
±
b
m
2
+
a
2
x
c
o
s
θ
b
+
y
s
i
n
θ
a
=
1
\frac{xcos\theta}{b}+\frac{ysin\theta}{a}=1
b
x
c
o
s
θ
+
a
y
s
i
n
θ
=
1
c
=
±
b
m
2
+
a
2
c=±\sqrt{bm^2+a^2}
c
=
±
b
m
2
+
a
2
Equations of Normal of Ellipse
E
q
u
a
t
i
o
n
Equation
E
q
u
a
t
i
o
n
P
a
r
a
m
e
t
r
i
c
C
o
o
r
d
i
n
a
t
e
s
Parametric\space Coordinates
P
a
r
a
m
e
t
r
i
c
C
o
o
r
d
i
n
a
t
e
s
E
q
u
a
t
i
o
n
o
f
N
o
r
m
a
l
Equation\space of\space Normal
E
q
u
a
t
i
o
n
o
f
N
o
r
m
a
l
C
o
n
d
i
t
i
o
n
o
f
N
o
r
m
a
l
i
t
y
Condition\space of \space Normality
C
o
n
d
i
t
i
o
n
o
f
N
o
r
m
a
l
i
t
y
x
2
a
2
+
y
2
b
2
=
1
\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1
a
2
x
2
+
b
2
y
2
=
1
a
>
b
a>b
a
>
b
(
a
c
o
s
θ
,
b
s
i
n
θ
)
(acos\theta,bsin\theta)
(
a
c
o
s
θ
,
b
s
i
n
θ
)
a
x
c
o
s
θ
−
b
y
s
i
n
θ
=
a
2
−
b
2
\frac{ax}{cos\theta}-\frac{by}{sin\theta}=a^2-b^2
c
o
s
θ
a
x
−
s
i
n
θ
b
y
=
a
2
−
b
2
c
=
±
m
(
a
2
−
b
2
)
a
2
+
b
2
m
2
c=±\frac{m(a^2-b^2)}{\sqrt{a^2+b^2m^2}}
c
=
±
a
2
+
b
2
m
2
m
(
a
2
−
b
2
)
x
2
a
2
+
y
2
b
2
=
1
\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1
a
2
x
2
+
b
2
y
2
=
1
a
<
b
a < b
a
<
b
(
b
c
o
s
θ
,
a
s
i
n
θ
)
(bcos\theta,asin\theta)
(
b
c
o
s
θ
,
a
s
i
n
θ
)
b
x
c
o
s
θ
−
a
y
s
i
n
θ
=
b
2
−
a
2
\frac{bx}{cos\theta}-\frac{ay}{sin\theta}=b^2-a^2
c
o
s
θ
b
x
−
s
i
n
θ
a
y
=
b
2
−
a
2
c
=
±
m
(
b
2
−
a
2
)
b
2
+
a
2
m
2
c=±\frac{m(b^2-a^2)}{\sqrt{b^2+a^2m^2}}
c
=
±
b
2
+
a
2
m
2
m
(
b
2
−
a
2
)
Equations of Director circle of Ellipse
E
q
u
a
t
i
o
n
Equation
E
q
u
a
t
i
o
n
E
q
u
a
t
i
o
n
o
f
D
i
r
e
c
t
o
r
C
i
r
c
l
e
Equation\space of\space Director\space Circle
E
q
u
a
t
i
o
n
o
f
D
i
r
e
c
t
o
r
C
i
r
c
l
e
x
2
a
2
+
y
2
b
2
=
1
\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1
a
2
x
2
+
b
2
y
2
=
1
a
>
b
a>b
a
>
b
x
2
+
y
2
=
a
2
+
b
2
x^2\,+\,y^2\space=\,a^2\,+\,b^2
x
2
+
y
2
=
a
2
+
b
2
x
2
a
2
+
y
2
b
2
=
1
\frac{x^2}{a^2}\space+\frac{y^2}{b^2}\space=1
a
2
x
2
+
b
2
y
2
=
1
a
<
b
a < b
a
<
b
x
2
+
y
2
=
a
2
+
b
2
x^2\,+\,y^2\space=\,a^2\,+\,b^2
x
2
+
y
2
=
a
2
+
b
2
