## 2.25. Cartesian Equations For The Ellipse And The
Hyperbola in Standard Position

Consider again the central conics

‖ X ‖
2
−
e
2
(
X⋅
N
ˆ
)
2
=
a
2
(
1−
e
2
)
(2.37)

Setting
X=(
x,y
)
and
N
ˆ
=(
1,0
)
,
we have

x
2
+
y
2
−
e
2
x
2
=
a
2
(
1−
e
2
)

Þ
x
2
a
2
+
y
2
a
2
(
1−
e
2
)
=1
(2.38)

which is called the **standard form** of the central conics. The foci are at
±ae
N
ˆ
=(
±ae,0
)
. The directrices are vertical lines
intersecting the *x*-axis at points
±
a
e
N
ˆ
=(
±
a
e
,0
)
.

For an ellipse with
e<1
,
we set
b=a
1−
e
2
so that (2.38) becomes

x
2
a
2
+
y
2
b
2
=1
(2.39)

and

c=ae
=a
1−
b
2
a
2
=
a
2
−
b
2

For a hyperbola with
e>1
,
we set
b=| a |
e
2
−1
so that (2.38) becomes

x
2
a
2
−
y
2
b
2
=1
(2.40)

and

c=| a |e
=| a |
1+
b
2
a
2
=
a
2
+
b
2

Now, as
| x |→∞
,
(2.40) becomes

y
2
b
2
≃
x
2
a
2
or
y≃±
b
| a |
x

These give 2 lines with tangents
±
b
| a |
that pass through the origin. They are called **asymptotes** of the hyperbola.
See Fig.2.14.