Handbook of mathematics for engineers and scienteists part 20 pot

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 equa

4.4.2-6 Ellipse in polar coordinate system.

In polar coordinates (ρ, ϕ), the equation of an ellipse becomes

ρ= p

where0 ≤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

x2

a2 –

y2

b2 =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.

M x y( , )

2 1

2 1

b

φ

b

Y

M

N

A

Y

Figure 4.22 Hyperbola.

The hyperbola is a central curve of second order It is described by equation (4.4.3.1)

and consists of two connected parts (arms) lying in the domains x > a and x < –a The hyperbola has two asymptotes given by the equations

y= b

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

tanϕ

2 =

b

and if a = b, then ϕ = 12π (an equilateral hyperbola).

The number a is called the real semiaxis, and the number b is called the imaginary

semiaxis The number c = √

a2+ b2 is called the linear eccentricity, and2cis called the

focal distance The number e = c/a = √

a2+ b2/a , where, obviously, e > 1, is called

the eccentricity, or the numerical eccentricity The number p = b2/a is called the focal

parameter or simply the parameter of the hyperbola.

The point O(0,0) is called the center of the hyperbola The points A1(–a,0) and A2(a,0)

of intersection of the hyperbola with the real axis are called the vertices of the hyperbola Points F1(–c,0) and F2(c,0) are called the foci of the hyperbola This is why the real axis of

a hyperbola is sometimes called the focal axis The straight lines x = a/e (y≠ 0) are called

the directrices of the hyperbola corresponding to the foci F2and F1 The focus F2(c,0) and

the directrix x = a/e are said to be right, and the focus F1(–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 F1(–c,0) and

F2(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 r1=|F1M| and r2=|F2M|, 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 x2– y2 = a2 If we take the asymptotes to be the

coordinate axes, then the equation of the hyperbola becomes xy = a2/2 ; i.e., an equilateral hyperbola is the graph of inverse proportionality.

The curvature radius of a hyperbola at a point M (x, y) is

R = a2b2

x2

a4 +

y2

b4

3 2

=

(r1r2)3

ab (4.4.3.4)

The area of the figure bounded by the right arm of the hyperbola and the chord passing

through the points M (x1, y1) and N (x1, –y1) is equal to (see Fig 4.22b)

S = x1y1– ab ln

x1

a + y1

b



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 F1and F2has the same absolute value2a

(see Fig 4.22a) We write this property as

|r1– r2|=2a, (4.4.3.6)

where r1and r2satisfy the relations

r1=

(x + c)2+ y2 =

a + ex for x >0,

–a – ex for x <0,

r2=

(x – c)2+ y2 =



–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.

4.4.3-3 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:

r1x + a e



–1= e, r2x – a

e



–1= e. (4.4.3.8)

4.4.3-4 Equation of tangent and optical property of hyperbola

The tangent to the hyperbola (4.4.3.1) at an arbitrary point M0(x0, y0) is described by the equation

x0x

a2 –

y0y

b2 =1 (4.4.3.9)

The distances d1 and d2 from the foci F1(–c,0) and F2(0, c) to the tangent to the hyperbola at the point M0(x0, y0) are given by the formulas (see Paragraph 4.3.2-4)

d1= N a

|x0e + a| =

r1

N a,

d2= |x0N a e – a| =

r2

N a,

N = x0

a2

2 +

y

0

b2

2 , (4.4.3.10)

where r1and r2are the lengths of the focal radii of the point M0

M

F

r

d

d r

1

1

2 2 2 1

2

0

1

φ φ

Y

Figure 4.23 The tangent to the hyperbola (a) Optical property of a hyperbola (b).

The tangent at any point M0(x0, y0) of the hyperbola forms acute angles ϕ1and ϕ2with

the focal radii of the point of tangency (see Fig 4.23a), and

sin ϕ1 = d1

r1 =

1

N a, sin ϕ2= d2

r2 =

1

N a (4.4.3.11)

This implies the optical property of a hyperbola:

ϕ1= ϕ2, (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 to2b

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

k1k2 = b

2

a2 . (4.4.3.13)

A hyperbola meets the diameter y = kx if and only if

k2< b2

a2 . (4.4.3.14)

The lengths l1and l2of the conjugate diameters with slopes k1and k2satisfy the relation

l1l2sin(arctan k2– arctan k1) =4ab (4.4.3.15)

Two perpendicular conjugate diameters are called the principal diameters of a

hyper-bola; they are its principal axes.

4.4.3-6 Hyperbola in polar coordinate system

In polar coordinates (ρ, ϕ), the equation for two connected parts of a hyperbola becomes

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

y2=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

X

p

r

Y

M

Y

M

N

Figure 4.24 Parabola.

A parabola is a noncentral line of second order It consists of an infinite branch

symmetric about the OX-axis The point O(0,0) is called the vertex of the parabola The point F (p/2,0) is called the focus of the parabola The straight line x = –p/2 is called

the directrix The focal parameter p is the distance from the focus to the directrix The number p/2is called the focal distance.

The segment joining a point M (x, y) of the parabola with the focus F (p/2,0) is called

the focal radius of the point The curvature radius of the parabola at a point M (x, y) can be

found from the formula

R= (p +2x)3 2

√

The area of the figure bounded by the parabola and the chord passing through the

points M (x1, y1) and N (x1, –y1) is equal to (see Fig 4.24b)

S= 4

3x1y1. (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

where r satisfies the relation

r = x– p

2

2

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 to1:

r

|x+ p/2| =1. (4.4.4.6)

4.4.4-4 Equation of tangent and optical property of parabola

The tangent to the parabola (4.4.4.1) at an arbitrary point M0(x0, y0) is described by the equation

The direction vector of the tangent (4.4.4.7) has the coordinates (y0, p), and the direction vector of the line passing through the points M0(x0, y0) and F (p/2,0) has the coordinates

O X

φ φ

F

M0

Y

Figure 4.25 The tangent to the parabola (a) Optical property of a parabola (b).

(x0– p/2, y0) (see Fig 4.25a) Thus, in view of the focus-directrix property, the angle ϕ

between these lines satisfies the relation

cos ϕ = y0(x0– p/2) + py0

y2

0+ p2

(x0– p/2)2+ y20 =

y0

y2

0+ p2

(4.4.4.8)

But the same relation also holds for the angle between the tangent (4.4.4.7) and the OX-axis This property of a parabola is called the optical property: all light rays issuing from

the focus of a parabola form a pencil parallel to the axis of the parabola after the mirror

reflection in the parabola (see Fig 4.25b).

The tangent and normal to a parabola at any point bisect the angles between the focal radius and the diameter

4.4.4-5 Diameters of parabola

A straight line passing through the midpoints of parallel chords of a parabola is called a

diameter of the parabola The diameter corresponding to the chords perpendicular to the

axis of the parabola is the axis itself The diameter of the parabola y2=2pxcorresponding

to the chords with slope k (k >0) is given by the equation

y= p

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 = ax2+ bx + c (a≠ 0) (4.4.4.10)

For a >0, the vertex of the parabola is directed downward, and for a < 0, the vertex is directed upward The vertex of a parabola has the coordinates

x0= b2, y0=

4ac– b2

4.4.4-7 Parabola in polar coordinates.

In the polar coordinates (ρ, ϕ) (the pole lies at the focus of the parabola, and the polar axis

is directed along the parabola axis), the equation of the parabola has the form

ρ= p

where –12π≤ϕ≤ 1

2π.

4.4.5 Transformation of Second-Order Curves to Canonical Form

4.4.5-1 General equation of second-order curve

The set of points on the plane whose coordinates in the rectangular Cartesian coordinate system satisfy the second-order algebraic equation

a11x2+2a12xy + a22y2+2a13x+2a23y + a33 =0 or

(a11x + a12y + a13)x + (a21x + a22y + a23)y + a31x + a32y + a33=0,

a ij = a ji (i, j =1,2,3)

(4.4.5.1)

is called a second-order curve.

4.4.5-2 Nine canonical second-order curves

There exists a rectangular Cartesian coordinate system in which equations (4.4.5.1) can be

reduced to one of the following nine canonical forms:

1 x

2

a2 +

y2

b2 =1, an ellipse;

2 x

2

a2 –

y2

b2 =1, a hyperbola;

3 y =2px, a parabola;

4 x

2

a2 +

y2

b2 = –1, an imaginary ellipse;

5 x

2

a2 –

y2

b2 =0, a pair of intersecting straight lines;

6 x

2

a2 +

y2

b2 =0, a pair of imaginary intersecting straight lines;

7 x2– a2=0, a pair of parallel straight lines;

8 x2+ a2=0, a pair of imaginary parallel straight lines;

9 x2=0, a pair of coinciding straight lines

4.4.5-3 Invariants of second-order curves

Second-order curves can be studied with the use of the three invariants

I = a11+ a22, δ=a22 a23

a32 a33



 , Δ =





a11 a12 a13

a21 a22 a23

a31 a32 a33





, (4.4.5.2)

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... 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... data-page="4">

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