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ANALYTIC GEOMETRY

Vocabulary

angular relations – угловые соотношения asymptote - асимптота

axis of the parabola - ось параболы

canonical equations – канонические уравнения directrix - директриса

distance from a point to – расстояние от точки до eccentricity – эксцентриситет

ellipse - эллипс

ellipse’s focal axis – фокальные оси эллипса ellipsoid - эллипсоид

elliptic cone – эллиптический конус elliptic cylinder – эллиптический цилиндр

elliptic paraboloid – эллиптический параболоид general equation – общее уравнение

hyperbolic cylinder – гиперболический цилиндр hyperbolic paraboloid – гиперболический параболоид

intercept form of the equation of a plane – уравнение плоскости в отрезках major axis – большая ось

normal vector - нормальный вектор

one-sheeted hyperboloid – однополостный гиперболоид parabola – парабола

parabolic cylinder – параболический цилиндр position vector - направляющий вектор quadric surface – поверхность второго порядка second order curves – кривые второго порядка semimajor axis - большая полуось

slope equation - уравнение с угловым коэффициентом standard parametric equations – параметрические уравнения

two-points equation of a straight line – уравнение прямой через две точки two-sheeted hyperboloid – двуполостный гиперболоид

vector equation – векторное уравнение vertex of the parabola – вершина параболы

№

Names, definitions, theorems

Equations and formulas

A plane in a space

The equation for the plane through the

A(x − x0 ) + B(y − y0 ) + C(z − z0 ) = 0

point M (x0 ; y0 ; z0 ) normal to the

vector

= (A, B, C )

Standard equation of a plane π

π : Ax + By + Cz + D = 0

= (A; B; C ) π

Equation of a plane passing through

x − x1

y − y1

z − z1

points P (x ; y ; z ),

P (x

; y

; z ),

x2 − x1

y2 − y1

z2 − z1

= 0

P3 (x3 ; y3 ; z3 ).

x3 − x1

y3 − y1

z3 − z1

Intercept form of the equation of a

= 1

plane

b c

5The distance from the point

P(x0 ; y0 ; z0 ) to the

plane

d (P,π ) =

Ax0 + By0 + Cz0 + D

π : Ax + By + Cz + D = 0

A2 + B 2 + C 2

Let

1 = (A1 ; B1 ; C1 ) - normal vector of a plane π1 , аnd

2 = (A2 ; B2 ; C2 ) - normal

vector of a plane π 2 , then there are relations:

Conditions of the perpendicularity

of the planes: π1 π 2

π1 π 2 A1 A2 + B1 B2 + C1C2

Conditions of the parallelism of

π1

π 2 A1

= B1 = C1

the planes: π1

π 2

B2 C2

The angle between

the planes

cos (π1 ^π 2 ) =

cos (

1 , ^

2 )

(π1 ^π 2 )

A straight line in a space

Let a straight line lpasses the point M (x0 ; y0 ; z0 ) parallel to a vector s = (m; n; p),

(

− directed vector of l)

then:

Vector equation of a

straight line l

r = r0

+ t s

l :

x − x

y − y

z − z

Canonical equations of a straight line l

		x = x0	+ mt
11	Standard parametric equations of a	l : y = y0	+ nt
	straight line l	z = z0	+ pt
		z = z0	+ pt

Let s1 = (m1 ; n1 ; p1 )- direction vector of a straight line l1 , аnd s2 direction vector of a straight line l 2 , then there are relations:

= (m2 ; n2 ; p2 ) -

Conditions of the parallelism of the

straight lines: l1

l 2

Conditions of the

perpendicularity of

l1 l 2 m1m2 + n1n2 + p1 p2 = 0

the straight lines:

l1 l 2

The angle

between

the straight

cos (l1^ l 2 ) =

m1m2 + n1n2 + p1 p2

lines: cos (l

^ l

) =

cos (

, ^

)

m 2

+ n 2

+ p 2

m 2 + n

+ p

A plane and a straight line in a space

Let l :

x − x0

y − y0

z − z0

and π : Ax + By + Cz + D = 0 be given, then:

Parallelism of a straight line l and a

l Am + Bn + Cp = 0

plane π : π

π l

Conditions of the perpendicularity of

m n

a straight line l and a plane π :

π l

Angle ϕ between a straight line l

sin (π ^ l) =

and a plane π :

Am + Bn + Cp

sin (π ^ l) =

cos (n, ^ s )

A2 + B 2 + C 2

m 2 + n 2 + p 2

A straight line in a plane

The first group of equations of a straight line (with normal vector n = (A, B))

The equation of a straight line

A(x − x0 ) + B(y − y0 ) = 0

through the point P0 (x0 ; y0 ) with the

normal vector

= (A; B).

A general equation of a straight

Ax + By + C = 0

line

A distance from a point P0 (x0 ; y0 ) to a

d =

Ax0 + By0 + C

straight line Ax + By + C = 0

A2 + B 2

Intercept form of the equation of a

= 1

straight line

a b

The second group of equations of a straight line (with direction vector s = (m, n))

A vector equation

r = r0 + t s ,

x = x0 + mt

A parametric equations

+ nt

y = y0

x − x0

y − y0

A canonical equation

x − x1

y − y1

A two-points equation

x2 − x1

y2 − y1

	The third group of equations of a straight line in a plane:
	the straight line with a slope ( k = tan ϕ , ϕ = l Ox )

26	the point – slope equation of a straight	y − y0 = k (x − x0 )
	line
27	the slope-intercept equation of a straight	y = kx + b
	line
	44

Angular relations between straight lines l1 and l 2

A1	=	B1

A2		B2

Conditions of parallelism of straight lines

l 2

28 l1 and l 2

= k

Conditions of

the

perpendicularity of

lines l1 and l 2

A1 A2 + B1 B2 = 0

l1 l

2 m1m2 + n1n2 = 0

= −

k 2

Finding an angle θ between straight lines

A1 A2 + B1 B2

l1 and l 2 :

cosθ =

A12

+ B12

A22 + B22

(

)

cosθ

cos n1

m1m2 + n1n2

(

)

cosθ

cos s1

+ n1

+ n2

tan ϕ1 − tan ϕ 2

− k

tanθ =

tanθ

1 + tan ϕ1 tan ϕ 2

1 + k

SECOND ORDER CURVES

Parabolas

x 2 = 4 py

y 2 = 4 px

y						y	y2 = 4 px
y						p	y2 = 4 px
			x2			p
F(0, p)	y =		x2			=
F(0, p)	y =		4p			=
						x
			P(x,	y)				x
				x	Elli			x
				x		0
	0					0	F (0,	p)
			Q (x,	p)	pse:		F (0,	p)
			Q (x,	p)	pse:
	Directrix	y = p
						45

x 2	+	y		2
	+			= 1
a 2		b		2
					y
				b					P(x, y)
									P(x, y)
										x
	F		(	c, 0)	0	F	( ,	)		a
			(	c, 0)	0		( ,	)
							c	0

foci F1 (− c,0) and F2 (c,0) , where c = a 2 − b 2

a 2 − b 2

eccentricity: e =

Hyperbola:		x 2	−	y	2
Hyperbola:			−		= 1
		a 2		b	2
							bx
		y			y	=		a
						=

	b				P(x , y)
					x
F ( c, 0)	a	0	a		F(c, 0)
F ( c, 0)		0			F(c, 0)
	b
					y		=		b
							=		b	x
									a	x

1.Planes

1)A General Equation of a Plane

Ax + By + Cz + D = 0

Foci F1 (− c,0) and F2 (c,0), where

c = a 2 + b 2
Eccentricity:	e =	c	=	a 2	+ b 2
Eccentricity:	e =	a	=		a
		a			a

Asymptotes y = ± b x

	z
D
C
	D
O	B
O

D	y
A

2) Planes which are Parallel to One of the Coordinat Axis

a) π1		Oy	b)π 2		Oz	c)π 3		Ox
π1 :Ax + Cz + D = 0			π 2 :Ax + By + D = 0			π 3 :By + Cz + D = 0

	z			z
	z		D
D		z	D
D		z	C
C			C
C
		D		D
		D		B
O	O	B	O	B
		B

D	y	y		y
A

x	x	D	x
	x	A
		A

3) Planes which are Passing through One of the Axis of Coordinates

a) Oy π 1	b)π 2 Oz	c)π 3 Ox
π1 :Ax + Cz = 0	π 2 :Ax + By = 0	π 3 :By + Cz = 0

z	z
O	O
y	y
x	x

4) Planes which are Parallel to One of the Coordinat Planes

a) π 1

xOz

b)π 2

yOz

c)π 3

xOy

π1 : y = D

π 2 : x = D

π 3 : z = D

2. Quadric Surfaces

Elliptic Cylynder

х	2	+	у	2	= 1

а2			b2
а2			b2

Elliptic Cone

х2 + у2 − z 2 = 0 а2 b2 c2

Hyperbolic Cylinder

х2 − у2 = 1

а2 b2

z
b	y
b
a
x

Parabolic Cylinder

x 2 = 2 py

Hyperbolic Paraboloid

х2 − у2 = 2z

Elliptic Paraboloid

х2	+ у2	= z
а2	b2
		z
		b
		a
		y
		x

Two-sheeted Hyperboloid

х2 + у2 − z 2 = −1

а2 b2 c2

	z
a	b
	y
x

One-sheeted Hyperboloid

Ellipsoid

х2	у2	z 2	х2	+	у2	+	z 2	= 1
+	−	= 1	а2	+	b2	+	c2	= 1
а2	b2	c2	а2		b2		c2
		z
		b		z
	a	b
	a
c				c
			y	c				y
			y				b

	x		a
	x		a
			x

LIMITS

Vocabulary approach x0 - достигать x0

arbitrarily number – произвольное число

comparison of infinitesimals – сравнение бесконечно малых deleted neighborhood – выколотая окрестность

equivalent infinitesimals – эквивалентные бесконечно малые however small – как угодно малый

indeterminacy - неопределенность

infinitely large function – бесконечно большая функция infinitesimal – бесконечно малая

infinitesimal of higher order – бесконечно малая более высокого порядка infinitesimal of the same order – бесконечно малые одного порядка left-hand limit - левосторонний предел

limit of a sequence – предел последовательности neighborhood – окрестность

one-sided limits – односторонние пределы reciprocal – обратный по величине remarkable limit – замечательный предел right-hand limit - правосторонний предел strip - полоса

tend to x0 - стремиться к x0

Definition of a Limit of a Function

lim f (x ) = L : L is the limit of the function

f (x)

x→ x0

f(x) lies in here

as x tends to x0 (approaches x0 )

1. Remember: lim f (x) = f (x0 )

x→ xo

x0 D

(x0 ):

for all x=x0 in here

lim f (x ) = L ε > 0 U δ

x→ x0

f (x ) − L

< ε

x U δ (x0 )

where

U δ (x0 )is deleted neighborhood of a point x0 .

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х2	у2	z 2	х2	+	у2	+	z 2	= 1
+	−	= 1	а2	+	b2	+	c2	= 1
а2	b2	c2	а2		b2		c2
		z
		b		z
	a	b
	a
c				c
			y	c				y
			y				b

	x		a
	x		a
			x

х2	у2	z 2	х2	+	у2	+	z 2	= 1
+	−	= 1	а2	+	b2	+	c2	= 1
а2	b2	c2	а2		b2		c2
		z
		b		z
	a	b
	a
c				c
			y	c				y
			y				b

	x		a
	x		a
			x

х2	у2	z 2	х2	+	у2	+	z 2	= 1
+	−	= 1	а2	+	b2	+	c2	= 1
а2	b2	c2	а2		b2		c2
		z
		b		z
	a	b
	a
c				c
			y	c				y
			y				b

	x		a
	x		a
			x