4.4. SECOND-ORDER CURVES 101
4.4.2-6. Ellipse in polar coordinate system.
In polar coordinates (ρ, ϕ), the equation of an ellipse becomes
ρ =
p
1 – e cos ϕ
,(4.4.2.11)
where 0 ≤ ϕ ≤ 2π.
4.4.3. Hyperbola
4.4.3-1. Definition and canonical equation of hyperbola.
A curve on the plane is called a hyperbola if there exists a rectangular Cartesian coordinate
system OXY in which the equation of this curve has the form
x
2
a
2
–
y
2
b
2
= 1,(4.4.3.1)
where a > 0 and b > 0 (see Fig. 4.22a). The coordinates in which the equation of a
hyperbola has the form (4.4.3.1) are called the canonical coordinates for the hyperbola, and
equation (4.4.3.1) itself is called the canonical equation of the hyperbola.
O
X
F
Mxy(,)
F
A

x and y =–
b
a
x.(4.4.3.2)
More precisely, its arms lie in the two vertical angles formed by the asymptotes and are
called the left and right arms of the hyperbola. A hyperbola is symmetric about the axes OX
and OY , which are called the principal (real, or focal, and imaginary) axes.
The angle between the asymptotes of a hyperbola is determined by the equation
tan
ϕ
2
=
b
a
,(4.4.3.3)
and if a = b,thenϕ =
1
2
π (an equilateral hyperbola).
102 ANALYTIC GEOMETRY
The number a is called the real semiaxis, and the number b is called the imaginary
semiaxis. The number c =
√
a
2
+ b
2
is called the linear eccentricity,and2c is called the
focal distance. The number e = c/a =
√

(–c, 0) and the directrix x =–a/e
are said to be left. A focus and a directrix are said to be like if both of them are right or left
simultaneously.
The segments joining a point M (x, y) of the hyperbola with the foci F
1
(–c, 0)and
F
2
(c, 0) are called the left and right focal radii of this point. We denote the lengths of the
left and right focal radii by r
1
= |F
1
M| and r
2
= |F
2
M|, respectively.
Remark. For a = b, the hyperbola is said to be equilateral, and its asymptotes are mutually perpendicular.
The equation of an equilateral hyperbola has the form x
2
– y
2
= a
2
. If we take the asymptotes to be the
coordinate axes, then the equation of the hyperbola becomes xy = a
2
/2; i.e., an equilateral hyperbola is the
graph of inverse proportionality.

, y
1
)andN(x
1
,–y
1
) is equal to (see Fig. 4.22b)
S = x
1
y
1
– ab ln

x
1
a
+
y
1
b

.(4.4.3.5)
4.4.3-2. Focal properties of hyperbola.
The hyperbola determined by equation (4.4.3.1) is the locus of points on the plane for
which the difference of the distances to the foci F
1
and F
2
has the same absolute value 2a
(see Fig. 4.22a). We write this property as

=

–a + ex for x > 0,
a – ex for x < 0.
(4.4.3.7)
Remark. One can show that equation (4.4.3.1) implies equation (4.4.3.6) and vice versa; hence the focal
property of a hyperbola is often used as the definition.
4.4. SECOND-ORDER CURVES 103
4.4.3-3. Focus-directrix property of hyperbola.
The hyperbola defined by equation (4.4.3.1) on the plane is the locus of points for which
the ratio of distances to a focus and the like directrix is equal to e:
r
1



x +
a
e



–1
= e, r
2



x –
a

2
from the foci F
1
(–c, 0)andF
2
(0, c) to the tangent to the
hyperbola at the point M
0
(x
0
, y
0
) are given by the formulas (see Paragraph 4.3.2-4)
d
1
=
Na
|x
0
e + a|
=
r
1
Na
,
d
2
=
Na
|x

0
.
O
X
F
M
F
r
d
d
r
1
1
2
2
2
1
2
0
1
φ
φ
Y
()a ()b
O
X
F
F
2
1

2
=
1
Na
.(4.4.3.11)
This implies the optical property of a hyperbola:
ϕ
1
= ϕ
2
,(4.4.3.12)
which means that all light rays issuing from a focus appear to be issuing from the other
focus after the mirror reflection in the hyperbola (see Fig. 4.23b).
The tangent and normal to a hyperbola at any point bisect the angles between the straight
lines joining this point with the foci. The tangent to a hyperbola at either of its vertices
intersects the asymptotes at two points such that the distance between them is equal to 2b.
104 ANALYTIC GEOMETRY
4.4.3-5. Diameters of hyperbola.
A straight line passing through the midpoints of parallel chords of a hyperbola is called a
diameter of the hyperbola. Two diameters of a hyperbola are said to be conjugate if their
slopes satisfy the relation
k
1
k
2
=
b
2
a
2

4.4.3-6. Hyperbola in polar coordinate system.
In polar coordinates (ρ, ϕ), the equation for two connected parts of a hyperbola becomes
ρ =
p
1 e cos ϕ
,(4.4.3.16)
where upper and lower signs correspond to right and left parts of a hyperbola, respectively.
4.4.4. Parabola
4.4.4-1. Definition and canonical equation of parabola.
A curve on the plane is called a parabola if there exists a rectangular Cartesian coordinate
system OXY , in which the equation of this curve has the form
y
2
= 2px,(4.4.4.1)
where p > 0 (see Fig. 4.24a). The coordinates in which the equation of a parabola has the
form (4.4.4.1) are called the canonical coordinates for the parabola, and equation (4.4.4.1)
itself is called the canonical equation of the parabola.
O
(a)(b)
X
p
2
F
r
Y
M
O
X
Y
M

x
1
y
1
.(4.4.4.3)
4.4.4-2. Focal properties of parabola.
The parabola defined by equation (4.4.4.1) on the plane is the locus of points equidistant
from the focus F (p/2, 0) and the directrix x =–p/2 (see Fig. 4.24a).
We denote the length of the focal radius by r and write this property as
r = x +
p
2
,(4.4.4.4)
where r satisfies the relation
r =


x –
p
2

2
+ y
2
.(4.4.4.5)
Remark. One can show that equation (4.4.4.1) implies equation (4.4.4.5) and vice versa; hence the focal
property of a parabola is often used as the definition.
4.4.4-3. Focus-directrix property of parabola.
The parabola defined by equation (4.4.4.1) on the plane is the locus of points for which the
ratio of distances to the focus and the directrix is equal to 1:

φφ
φ
F
Y
()a ()b
M
0
O
X
F
Y
Figure 4.25. The tangent to the parabola (a). Optical property of a parabola (b).
(x
0
– p/2, y
0
)(seeFig.4.25a). Thus, in view of the focus-directrix property, the angle ϕ
between these lines satisfies the relation
cos ϕ =
y
0
(x
0
– p/2)+py
0

y
2
0
+ p

= 2px corresponding
to the chords with slope k (k > 0) is given by the equation
y =
p
k
.(4.4.4.9)
The OX-axis (the axis of symmetry of a parabola), in contrast to the other diameters
of the parabola, is the diameter perpendicular to the chords conjugate to it. This diameter
is called the principal diameter of the parabola. The slope of any diameter of a parabola is
zero. A parabola does not have mutually conjugate diameters.
4.4.4-6. Parabola with vertical axis.
The equation of a parabola with vertical axis has the form
y = ax
2
+ bx + c (a ≠ 0). (4.4.4.10)
For a > 0, the vertex of the parabola is directed downward, and for a < 0,thevertexis
directed upward. The vertex of a parabola has the coordinates
x
0
=
b
2
, y
0
=
4ac – b
2
4a
.(4.4.4.11)
4.4. SECOND-ORDER CURVES 107

x + 2a
23
y + a
33
= 0 or
(a
11
x + a
12
y + a
13
)x +(a
21
x + a
22
y + a
23
)y + a
31
x + a
32
y + a
33
= 0,
a
ij
= a
ji
(i, j = 1, 2, 3)
(4.4.5.1)

a
2
+
y
2
b
2
=–1, an imaginary ellipse;
5.
x
2
a
2
–
y
2
b
2
= 0, a pair of intersecting straight lines;
6.
x
2
a
2
+
y
2
b
2
= 0, a pair of imaginary intersecting straight lines;

33



, Δ =





a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33

