80 ANALYTIC GEOMETRY OO O AA A x x x x y y y x X α α XX xx yy yy 0 0 0 0 YY Y Y X X X O O YY ()a ()b ()c Figure 4.3. Transformation of Cartesian coordinates under parallel translation (a), under rotation (b), and under translation and rotation (c)ofaxes. Let an arbitrary point A have coordinates (x, y) in the system OXY and coordi- nates (ˆx,ˆy) in the system O X Y . The transformation of rectangular Cartesian coordinates by the rotation of axes is given by the formulas x =ˆx cos α –ˆy sin α, y =ˆx sin α +ˆy cos α or ˆx = x cos α + y sin α, ˆy =–x sin α + y cos α. (4.1.2.3) 4.1.2-4. Transformation of coordinates under translation and rotation of axes. Suppose that two rectangular Cartesian coordinate systems OXY and O X Y are given and the first system is taken to the second by the translation of the origin O(0, 0)ofthefirst system to the origin O(x 0 , y 0 ) of the second system followed by the rotation of the system around the point O by an angle α (see Fig. 4.3c and Paragraphs 4.1.2-2 and 4.1.2-3). Let an arbitrary point A have coordinates (x, y) in the system OXY and coordi- nates (ˆx,ˆy) in the system O X Y . The transformation of rectangular Cartesian coordinates by the parallel translation and rotation of axes is given by the formulas x =ˆx cos α –ˆy sin α + x 0 , y =ˆx sin α +ˆy cos α + y 0 , or ˆx =(x – x 0 )cosα +(y – y 0 )sinα, ˆy =–(x – x 0 )sinα +(y – y 0 )cosα. (4.1.2.4) 4.1.2-5. Polar coordinates. A polar coordinate system is determined by a point O called the pole,arayOA issuing from this point, which is called the polar axis, a scale segment for measuring lengths, and the positive sense of rotation around the pole. Usually, the anticlockwise sense is assumed to be positive (see Fig. 4.4a). The position of each point B on the plane is determined by two polar coordinates,the polar radius ρ = |OB| and the polar angle θ = ∠AOB (the values of the angle θ are defined up to the addition of 2πn,wheren is an integer). To be definite, one usually assumes that 0 ≤ θ ≤ 2π or –π ≤ θ ≤ π. The polar radius of the pole is zero, and its polar angle does not have any definite value. 4.1.2-6. Relationship between Cartesian and polar coordinates. Suppose that B is an arbitrary point on the plane, (x, y) are its rectangular Cartesian coor- dinates, and (ρ, θ) are its polar coordinates (see Fig. 4.4b). The formulas of transformation 4.1. POINTS,SEGMENTS, AND COORDINATES ON LINE AND PLANE 81 O O A BB x X y Y ρρ θθ ()a ()b Figure 4.4. A polar coordinate system (a). Relationship between Cartesian and polar coordinates (b). from one coordinate system to the other have the form x = ρ cos θ, y = ρ sin θ or ρ = x 2 + y 2 , tan θ = y/x, (4.1.2.5) where the polar angle θ is determined with regard to the quadrant where the point B lies. Example 2. Let us find the polar coordinates ρ, θ (0 ≤ θ ≤ 2π) of the point B whose Cartesian coordinates are x =–3, y =–3. From formulas (4.1.2.5), we obtain ρ = (–3) 2 +(–3) 2 = 3 √ 2 and tan θ = –3 –3 = 1. Since the point B lies in the third quadrant, we have θ =arctan1 + π = 5 4 π. 4.1.3. Points and Segments on Plane 4.1.3-1. Distance between points on plane. The distance d between two arbitrary points A 1 and A 2 on the plane is given by the formula d = (x 2 – x 1 ) 2 +(y 2 – y 1 ) 2 ,(4.1.3.1) where x and y with the corresponding subscripts are the Cartesian coordinates of these points, and by the formula d = ρ 2 1 + ρ 2 2 – 2ρ 1 ρ 2 cos(θ 2 – θ 1 ), (4.1.3.2) where ρ and θ with the corresponding subscripts are the polar coordinates of these points. 4.1.3-2. Segment and its projections. Suppose that an axis u and an arbitrary segment −−→ A 1 A 2 are given on the plane (see Fig. 4.5a). From the points A 1 and A 2 , we draw the perpendiculars to u and denote the points of intersection of the perpendiculars with the axis by P 1 and P 2 .ThevalueP 1 P 2 of the segment −−→ P 1 P 2 of the axis u is called the projection of the segment −−→ A 1 A 2 onto the axis u. Usually one writes pr u −−→ A 1 A 2 = P 1 P 2 .Ifϕ (0 ≤ ϕ ≤ π) is the angle between the segment −−→ A 1 A 2 and the axis u,then pr u −−→ A 1 A 2 = d cos ϕ.(4.1.3.3) For two arbitrary points A 1 (x 1 , y 1 )andA 2 (x 2 , y 2 ), the projections x and y of the segment −−→ A 1 A 2 onto the coordinate X-andY-axes are given by the formulas (see Fig. 4.5b) pr X −−→ A 1 A 2 = x 2 – x 1 ,pr Y −−→ A 1 A 2 = y 2 – y 1 .(4.1.3.4) 82 ANALYTIC GEOMETRY O A A A A PP X x u d θ φ x y y 12 2 2 1 1 12 1 2 Y ()a ()b Figure 4.5. Projection of the segment onto the axis u (a) and onto the coordinate X-andY-axes (b). Thus, to obtain the projections of a segment onto the coordinate axes, one subtracts the coordinates of its initial point from the respective coordinates of its endpoint. The projections of the segment −−→ A 1 A 2 onto the coordinate axes can be found if its length d (see (4.1.3.1)) and polar angle θ are known (see Fig. 4.5b). The corresponding formulas are pr X −−→ A 1 A 2 = d cos θ,pr Y −−→ A 1 A 2 = d sin θ,tanθ = y 2 – y 1 x 2 – x 1 .(4.1.3.5) 4.1.3-3. Angles between coordinate axes and segments. The angles α x ≡ θ and α y between the segment −−→ A 1 A 2 and the coordinate x-andy-axes are determined by the expressions cos α x = x 2 – x 1 (x 2 – x 1 ) 2 +(y 2 – y 1 ) 2 ,cosα y = y 2 – y 1 (x 2 – x 1 ) 2 +(y 2 – y 1 ) 2 ,(4.1.3.6) and α y = π – α x . The angle β between arbitrary segments −−→ A 1 A 2 and −−→ A 3 A 4 joining the points A 1 (x 1 , y 1 ), A 2 (x 2 , y 2 )andA 3 (x 3 , y 3 ), A 4 (x 4 , y 4 ), respectively, can be found from the relation cos β = (x 2 – x 1 )(x 4 – x 3 )+(y 2 – y 1 )(y 4 – y 3 ) (x 2 – x 1 ) 2 +(y 2 – y 1 ) 2 (x 4 – x 3 ) 2 +(y 4 – y 3 ) 2 .(4.1.3.7) 4.1.3-4. Division of segment in given ratio. The number λ =p/q,wherep=A 1 A and q =AA 2 are the values of the directed segments −−→ A 1 A and −→ AA 2 , is called the ratio in which point A divides the segment −−→ A 1 A 2 . It is independent of the sense of the segment (i.e., one could use the segment −−→ A 2 A 1 )andthescalesegment. The coordinates of the point A dividing the segment −−→ A 1 A 2 in a ratio λ are given by the formulas x = x 1 + λx 2 1 + λ = qx 1 + px 2 q + p , y = y 1 + λy 2 1 + λ = qy 1 + py 2 q + p ,(4.1.3.8) where –∞ ≤ λ ≤ ∞. For the coordinates of the midpoint of the segment −−→ A 1 A 2 ,wehave x = x 1 + x 2 2 , y = y 1 + y 2 2 ;(4.1.3.9) i.e., each coordinate of the midpoint of a segment is equal to the half-sum of the respective coordinates of its endpoints. 4.1. POINTS,SEGMENTS, AND COORDINATES ON LINE AND PLANE 83 4.1.3-5. Area of triangle area. The area S 3 of the triangle with vertices A 1 , A 2 ,andA 3 is given by the formula S 3 = 1 2 [(x 2 – x 1 )(y 3 – y 1 )–(x 3 – x 1 )(y 2 – y 1 )] = 1 2 x 2 – x 1 y 2 – y 1 x 3 – x 1 y 3 – y 1 = 1 2 x 1 y 1 1 x 2 y 2 1 x 3 y 3 1 ,(4.1.3.10) where x and y with respective subscripts are the Cartesian coordinates of the vertices, and by the formula S 3 = 1 2 [ρ 1 ρ 2 sin(θ 2 – θ 1 )+ρ 2 ρ 3 sin(θ 3 – θ 2 )+ρ 3 ρ 1 sin(θ 1 – θ 3 )], (4.1.3.11) where ρ and θ with respective subscripts are the polar coordinates of the vertices. In formulas (4.1.3.10) and (4.1.3.11), one takes the plus sign if the vertices are numbered anticlockwise (see Fig. 4.6a) and the minus sign otherwise. OO A A A A A A A A XX 2 3 n-1 n 2 1 1 3 YY ()a ()b Figure 4.6. Area of triangle (a) and of a polygon (b). 4.1.3-6. Area of a polygon. The area S n of the polygon with vertices A 1 , , A n is given by the formula S n = 1 2 [(x 1 – x 2 )(y 1 + y 2 )+(x 2 – x 3 )(y 2 + y 3 )+···+(x n – x 1 )(y n + y 1 )], (4.1.3.12) where x and y with respective subscripts are the Cartesian coordinates of the vertices, and by the formula S n = 1 2 [ρ 1 ρ 2 sin(θ 2 – θ 1 )+ρ 2 ρ 3 sin(θ 3 – θ 2 )+···+ ρ n ρ 1 sin(θ 1 – θ n )], (4.1.3.13) where ρ and θ with respective subscripts are the polar coordinates of the vertices. In formulas (4.1.3.12) and (4.1.3.13), one takes the plus sign if the vertices are numbered anticlockwise (see Fig. 4.6b) and the minus sign otherwise. Remark. One often says that formulas (4.1.3.10)–(4.1.3.13) express the oriented area ofthe corresponding figures. 84 ANALYTIC GEOMETRY 4.2. Curves on Plane 4.2.1. Curves and Their Equations 4.2.1-1. Basic definitions. A curve on the plane determined by an equation in some coordinate system is the geometric locus of points of the plane whose coordinates satisfy this equation. An equation of a curve on the plane in a given coordinate system is an equation with two variables such that the coordinates of the points lying on the curve satisfy the equation and the coordinates of the points that do not lie on the curve do not satisfy it. The coordinates of an arbitrary point of a curve occurring in an equation of the curve are called current coordinates. 4.2.1-2. Equation of curve in Cartesian coordinate system. An equation of a curve in the Cartesian coordinate system OXY can be written as F (x, y)=0.(4.2.1.1) The image of a curve determined by an equation of the form y = f(x)(4.2.1.2) is called the graph of the function f (x). Example 1. Let us plot the curve determined by the equation x 2 – y = 0. We express one coordinate via the other (e.g., y via x) from this equation: y = x 2 . Specifying various values of x,wefind the corresponding values of y and thus construct a sequence of points of the desired curve. By joining these points, we obtain the curve itself (see Fig. 4.7a). O 4 4 8 12 16 4 O O 5 X x 4 3 1 1 3 2 0 2 4 y 16 9 1 1 9 4 0 4 16 X X Y Y y=x y 5=0 yx= yx= 2 Y ()a ()b ()c Figure 4.7. Cartesian coordinate system. Loci of points for equations x 2 – y = 0 (a), y – 5 = 0 (b), and x 2 – y 2 = 0 (c). 4.2.1-3. Special kinds of equations. 1. The equation of a curve on the plane may contain only one of the current coordinates but still determine a certain curve. 4.2. CURVES ON PLANE 85 Example 2. Suppose that the equation y –5 = 0 (or y = 5) is given. The locus of points whose coordinates are equal to five is the straight line parallel to the axis OX and passing through the point y = 5 of the axis OY (see Fig. 4.7b). Similarly, the equation x + 7 = 0 determines a straight line parallel to the axis OY . 2. If the left-hand side of equation (4.2.1.1) can be factorized, then, equating each factor separately with zero, we obtain several new equations, each of which can determine a certain curve. Example 3. Consider the equation x 2 – y 2 = 0. Factorizing the left-hand side of this equation, we obtain (x + y)(x – y)=0. Obviously, the latter equation determines the pair of straight lines x + y = 0 and x – y = 0, which are the bisectors of the coordinate angles (see Fig. 4.7c). 3. Equation (4.2.1.1) may determine a locus consisting of one or several isolated points. Example 4. The equation x 2 + y 2 = 0 determines the single point with coordinates (0, 0). Example 5. The equation (x 2 – 9) 2 +(y 2 – 25) 2 = 0 defines the locus consisting of the four points (3, 5), (3,–5), (–3, 5), and (–3,–5). 4. There exist equations that do not determine any locus. Example 6. The equation x 2 + y 2 + 5 = 0 does not have solutions for any real x and y. 4.2.1-4. Equation of curve in polar coordinate system. An equation of a curve in a polar coordinate system can be written as Φ(ρ, θ)=0,(4.2.1.3) where ρ is the polar radius and θ is the polar angle. This equation is satisfied by the polar coordinates of any point lying on the curve and is not satisfied by the coordinates of the points that do not lie on the curve. Example 7. Consider the equation ρ – a cos θ = 0 (or ρ = a cos θ), where a is a positive number. By B we denote the point with polar coordinates (ρ, θ), and by A we denote the point with coordinates (a, 0). If ρ = a cos θ, then the angle OBA is a right angle, and vice versa. Therefore, the locus of points whose coordinates satisfy this equation is a circle with diameter a (see Fig. 4.8). O θ A a ρ B Figure 4.8. Polar coordinate system. Locus of points for equations ρ – a cos θ = 0. Example 8. Consider the equation ρ – aθ = 0 (or ρ = aθ), where a is a positive constant. The curve determined by this equation is called a spiral of Archimedes. As θ increases starting from zero, the point B(ρ, θ) issues from the pole and moves around it in the positive sense, simultaneously moving away from it. For each point of this curve with positive coordinates (ρ, θ), one has the corresponding point (–ρ,–θ) on the same curve. Figures 4.9a and b show the branches of the spiral of Archimedes corresponding to the positive and negative values of θ, respectively. Note that the spiral of Archimedes divides each polar ray into equal segments (except for the segment nearest to the pole). 86 ANALYTIC GEOMETRY O O AA B B ()a θ >0 2aπ 2aπ 2aπ ()b θ <0 Figure 4.9. A spiral of Archimedes ρ = aθ corresponding to the positive (a) and negative (b)valuesofθ. O O a B ()a θ <0 θ >0 ρ θ ()b Figure 4.10. A hyperbolic spiral ρ = a/θ corresponding to the positive (a) and negative (b)valuesofθ. OO ()a p >1 0< <1p ()b Figure 4.11. A logarithmic spiral ρ = a θ corresponding to the values a > 1 (a)and0 < a < 1 (b). Example 9. Consider the equation ρ – a/θ = 0 (or ρ = a/θ), where a is a positive number. The curve determined by this equation is called a hyperbolic spiral. As θ increases, the point B(ρ, θ) moves around the pole in the positive sense while approaching it endlessly. As θ tends to zero, the point B approaches the line y = a while moving to infinity. Figures 4.10a and b show the branches of the hyperbolic spiral corresponding to the positive and negative values of θ, respectively. Example 10. Consider the equation ρ – a θ = 0 (or ρ = a θ ), where a is a positive number. The curve determined by this equation is called the logarithmic spiral. Figures 4.11a and b show the branches of the logarithmic spiral corresponding to the values a > 1 and 0 < a < 1, respectively. For a = 1, we obtain the equation of a circle. . perpendiculars to u and denote the points of intersection of the perpendiculars with the axis by P 1 and P 2 .ThevalueP 1 P 2 of the segment −−→ P 1 P 2 of the axis u is called the projection of the segment −−→ A 1 A 2 onto. rectangular Cartesian coor- dinates, and (ρ, θ) are its polar coordinates (see Fig. 4.4b). The formulas of transformation 4.1. POINTS,SEGMENTS, AND COORDINATES ON LINE AND PLANE 81 O O A BB x X y Y ρρ θθ ()a. α. (4.1.2.3) 4.1.2-4. Transformation of coordinates under translation and rotation of axes. Suppose that two rectangular Cartesian coordinate systems OXY and O X Y are given and the first system