If the straight lines are given by general equations 4.3.2.8, then a necessary and sufficient condition for them to be perpendicular can be written as see Paragraph 4.3.2-4 A1A2+ B1B2=0.
Trang 194 ANALYTICGEOMETRY
4.3.2-4 Angle between two straight lines
We consider two straight lines given by the equations
y = k1x + b1 and y = k2x + b2, (4.3.2.5)
where k1= tan ϕ1and k2= tan ϕ2are the slopes of the respective lines (see Fig 4.17) The
angle α between these lines can be obtained by the formula
tan α = k2– k1
1+ k1k2 , (4.3.2.6)
where k1k2≠–1 If k1k2= –1, then α = 12π
Remark. If at least one of the lines is perpendicular to the axis OX, then formula (4.3.2.6) does not make
sense In this case, the angle between the lines can be calculated by the formula
α = ϕ2 – ϕ1 (4 3 2 7 )
X
y=k x+b
y=k x+b
1
2
1
2
φ
α
φ Y
O
Figure 4.17 Angle between two straight lines.
The angle α between the two straight lines given by the general equations
A1x + B1y + C1 =0 and A2x + B2y + C2=0 (4.3.2.8) can be calculated using the expression
tan α = A1B2– A2B1
A1A2+ B1B2 , (4.3.2.9)
where A1A2+ B1B2≠ 0 If A1A2+ B1B2=0, then α = 12π
Remark If one needs to find the angle between straight lines and the order in which they are considered
is not defined, then this order can be chosen arbitrarily Obviously, a change in the order results in a change in the sign of the tangent of the angle.
Trang 24.3.2-5 Point of intersection of straight lines.
Suppose that two straight lines are defined by general equations in the form (4.3.2.8) Consider the system of two first-order algebraic equations (4.3.2.8):
A1x + B1y + C1 =0,
A2x + B2y + C2 =0 (4.3.2.10) Each common solution of equations (4.3.2.10) determines a common point of the tow lines
If the determinant of system (4.3.2.10) is not zero, i.e.,
A A1 B1
2 B2
= A1B2– A2B1≠ 0, (4.3.2.11) then the system is consistent and has a unique solution; hence these straight lines are distinct
and nonparallel and meet at the point A(x0, y0), where
x0= B1C2– B2C1
A1B2– A2B1 , y0=
C1A2– C2A1
A1B2– A2B1 . (4.3.2.12) Condition (4.3.2.11) is often written as
A1
A2 ≠ B1
Example 2 To find the point of intersection of the straight lines y =2x– 1and y = –4x+ 5 , we solve system (4.3.2.10):
2x – y –1 = 0 , – 4x – y +5 = 0 ,
and obtain x =1, y =1 Thus the intersection point has the coordinates ( 1 , 1 ).
4.3.2-6 Condition for straight lines to be perpendicular
For two straight lines determined by slope-intercept equations (4.3.2.5) to be perpendicular,
it is necessary and sufficient that
Relation (4.3.2.14) is usually written as
k1 = – 1
and one also says that the slopes of perpendicular straight lines are inversely proportional
in absolute value and opposite in sign
If the straight lines are given by general equations (4.3.2.8), then a necessary and sufficient condition for them to be perpendicular can be written as (see Paragraph 4.3.2-4)
A1A2+ B1B2=0 (4.3.2.16)
Example 3 The lines3x +y–3 = 0and x–3y+ 8 = 0 are perpendicular since they satisfy condition (4.3.2.16):
A1A2+ B1 B2 = 3 ⋅ 1 + 1 ⋅ (– 3 ) = 0
Trang 396 ANALYTICGEOMETRY
4.3.2-7 Condition for straight lines to be parallel
For two straight lines defined by slope-intercept equations (4.3.2.5) to be parallel and not
to coincide, it is necessary and sufficient that
k1 = k2, b1≠b2 (4.3.2.17)
If the straight lines are given by general equations (4.3.2.8), then a necessary and sufficient condition for them to be parallel can be written as
A1
A2 =
B1
B2 ≠ C1
in this case, the straight lines do not coincide (see Fig 4.18)
X b
b
1 2
φ φ Y
O
Figure 4.18 Parallel straight lines.
Example 4 The straight lines3x+ 4y+ 5 = 0 and 3/2x+ 2y+ 6 = 0 are parallel since the following condition (4.3.2.18) is satisfied:
3
3/2 =
4
2 ≠
5
6.
4.3.2-8 Condition for straight lines to coincide
For two straight lines given by slope-intercept equations (4.3.2.5) to coincide, it is necessary and sufficient that
k1 = k2, b1= b2. (4.3.2.19)
If the straight lines are given by general equations (4.3.2.8), then a necessary and sufficient condition for them to coincide has the form
A1
A2 =
B1
B2 =
C1
Remark Sometimes the case of coinciding straight lines is considered as a special case of parallel straight lines and it not distinguished as an exception.
4.3.2-9 Distance between parallel lines
The distance between the parallel lines given by equations (see Paragraph 4.3.2-7)
A1x + B1y + C1 =0 and A1x + B1y + C2=0 (4.3.2.21) can be found using the formula (see Paragraph 4.3.2-3)
d= |C1– C2|
A2
1+ B12
Trang 44.3.2-10 Condition for a straight line to separate points of plane.
Suppose that a straight line in the Cartesian coordinate system OXY is given by an equation
of the form
Obviously, this straight line divides the plane into two half-planes We consider two arbitrary
points A1(x1, y1) and A2(x2, y2) of the plane that do not lie on the line The points are said
to be nonseparated by the straight line if they belong to the same half-plane (lie on the same side of the straight line and possibly coincide) The points are said to be separated by
the straight line if they belong to different half-planes (lie on opposite sides of the straight
line)
Two points A1(x1, y1) and A2(x2, y2) that do not belong to the straight line (4.3.2.23)
are separated by this line if and only if the numbers Ax1+ By1+ C and Ax2+ By2+ C
have opposite signs
4.4 Second-Order Curves
4.4.1 Circle
4.4.1-1 Definition and canonical equation of circle
A curve on the plane is called a circle if there exists a rectangular Cartesian coordinate system OXY in which the equation of this curve has the form (see Fig 4.19a)
where the point O(0,0) is the center of the circle and a >0is its radius Equation (4.4.1.1)
is called the canonical equation of a circle.
M
N
a
a
Figure 4.19 Circle.
The circle defined by equation (4.4.1.1) is the locus of points equidistant (lying at the
distance a) from its center If a circle of radius a is centered at a point C(x0, y0), then its equation can be written as
(x – x0)2+ (y – y0)2= a2 (4.4.1.2)
The area of the disk bounded by a circle of radius a is given by the formula S = πa2
The length of this circle is L =2πa The area of the figure bounded by the circle and the
chord passing through the points M (x0, y0) and N (x0, –y0) is (see Fig 19b)
S = πa
2
2 + x0
a2– x2
0+ a2arcsinx a0 . (4.4.1.3)
Trang 598 ANALYTICGEOMETRY
4.4.1-2 Parametric and other equations of circle
The parametric equations of the circle (4.4.1.1) have the form
where the angle in the polar coordinate system plays the role of the variable parameter (see Paragraphs 4.2.1-4 and 4.2.1-5)
The equation of the circle (4.4.1.1) in the polar coordinate system has the form
and does not contain the polar angle θ.
Remark The form of the equation of a circle in a polar coordinate system depends on the choice of the pole and the polar axis (see Example 7 in Subsection 4.2.1).
4.4.2 Ellipse
4.4.2-1 Definition and canonical equation of ellipse
A curve on the plane is called an ellipse if there exists a rectangular Cartesian coordinate system OXY in which the equation of the curve has the form
x2
a2 +
y2
where a≥b>0(see Fig 4.20) The coordinates in which the equation of an ellipse has the
form (4.4.2.1) are called the canonical coordinates for this ellipse, and equation (4.4.2.1) itself is called the canonical equation of the ellipse.
M
N
b
b
Figure 4.20 Ellipse.
The segments A1A2 and B1B2 joining the opposite vertices of an ellipse, as well as their lengths2aand2b , are called the major and minor axes, respectively, of the ellipse The axes of an ellipse are its axes of symmetry In Fig 4.20a, the axes of symmetry of the ellipse coincide with the axes of the rectangular Cartesian coordinate system OXY The numbers a and b are called the semimajor and semiminor axes of the ellipse The number
c=√
a2– b2is called the linear eccentricity, and the number2c is called the focal distance The number e = c/a =
1– a2/b2, where, obviously,0 ≤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 ellipse.
Trang 6The point O(0,0) is called the center of the ellipse The points of intersection A1(–a,0),
A2(a,0) and B1(0, –b), B2(0, b) of the ellipse with the axes of symmetry are called its
the major axis of an ellipse is sometimes called its focal axis The straight lines x = a/e (e≠ 0) are called the directrices The focus F2(c,0) and the directrix x = a/e are said to be
directrix are said to be like if both of them are right or left simultaneously.
The segments joining a point M (x, y) of an ellipse 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=|F1M0|and r2=|F2M0|, respectively
Remark. For a = b (c =0), equation (4.4.2.1) becomes x2+ y2= a2and determines a circle; hence a circle
can be considered as an ellipse for which b = a, c =0, e =0, and ρ =0 , i.e., the semiaxes are equal to each other, the foci coincide with the center, the eccentricity is zero (the directrices are not defined), and the focal parameter is zero.
The area of the figure bounded by the ellipse is given by the formula S = πab The length
of the ellipse can be calculated approximately by the formula L≈π
1.5(a + b) – √
ab The
area of the figure bounded by the ellipse and the chord passing through the points M (x0, y0)
and N (x0, –y0) is equal to (see Fig 20b)
S= πab
2 +
b a
x0
a2– x2
0+ a2arcsin
x0
a
4.4.2-2 Focal property of ellipse
The ellipse defined by equation (4.4.2.1) is the locus of points on the plane for which the
sum of distances to the foci F1and F2is equal to2a(see Fig 4.21) We write this property as
where r1and r2satisfy the relations
r1 =
(x + c)2+ y2 = a + ex,
r2 =
(x – c)2+ y2= a – ex. (4.4.2.4)
2 2 1
2 1
Y
M0
2
φ
Figure 4.21 Focal property of ellipse.
Remark One can show that equation (4.4.2.1) implies equation (4.4.2.3) and vice versa; hence the focal property of an ellipse is often used as its definition.
Trang 7100 ANALYTICGEOMETRY
4.4.2-3 Focus-directrix property of ellipse
The ellipse determined by equation (4.4.2.1) 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. (4.4.2.5)
4.4.2-4 Equation of tangent and optical property of ellipse
The tangent to the ellipse (4.4.2.1) at an arbitrary point M0(x0, y0) is described by the equation
x0x
a2 +
y0y
The distances d1and d2from the foci F1(–c,0) and F2(c,0) to the tangent to the ellipse
at the point M0(x0, y0) are given by the formulas (see Paragraph 4.3.2-4)
d1 = 1
N a|x0e + a|= r1(M0)
N a ,
d2 = 1
N a|x0e – a|= r2(M0)
N a ,
a2
2 +
y0
b2
2 , (4.4.2.7)
where r1(M0) and r2(M0) are the lengths of the focal radii of M0
The tangent at an arbitrary point M0(x0, y0) of an ellipse forms acute angles ϕ1and ϕ2
with the focal radii of the point of tangency, and
sin ϕ1= d1
r1 =
1
N a, sin ϕ2= d2
r2 =
1
N a (4.4.2.8)
This implies the optical property of the ellipse:
which means that all light rays issuing from one focus of the ellipse converge at the other focus after the reflection in the ellipse
4.4.2-5 Diameters of ellipse
A straight line passing through the midpoints of parallel chords of an ellipse is called a
diameter of the ellipse All diameters of an ellipse pass through its center Two diameters
of an ellipse are said to be conjugate if their slopes satisfy the relation
k1k2= –b
2
Two perpendicular conjugate diameters are called the principal diameters of the ellipse.
Remark. If a = b, i.e., the ellipse is a circle, then condition (4.4.2.10) becomes the perpendicularity condition: k1 k2 = – 1 Thus any two conjugate diameters of a circle are perpendicular to each other, and each
of the diameters is a principal diameter.
... 4.2.1-4 and 4.2.1-5)The equation of the circle (4.4.1.1) in the polar coordinate system has the form
and does not contain the polar angle θ.
Remark The form of. .. The axes of an ellipse are its axes of symmetry In Fig 4.20a, the axes of symmetry of the ellipse coincide with the axes of the rectangular Cartesian coordinate system OXY The numbers a and b are... M0(x0, y0) of an ellipse forms acute angles ϕ1and ϕ2
with the focal radii of the point of tangency, and
sin ϕ1=