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 A r r 2 1 1 2 2 1 b ()a ()b φ b Y O X M N A A 2 1 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 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 = √ a 2 + b 2 /a, where, obviously, e > 1, is called the eccentricity,orthenumerical eccentricity. The number p = b 2 /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 A 1 (–a, 0)andA 2 (a, 0) of intersection of the hyperbola with the real axis are called the vertices of the hyperbola. Points F 1 (–c, 0)andF 2 (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 F 2 and F 1 .ThefocusF 2 (c, 0)and the directrix x = a/e are said to be right, and the focus F 1 (–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. The curvature radius of a hyperbola at a point M(x, y)is R = a 2 b 2 x 2 a 4 + y 2 b 4 3/2 = (r 1 r 2 ) 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(x 1 , 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 |r 1 – r 2 | = 2a,(4.4.3.6) where r 1 and r 2 satisfy the relations r 1 = (x + c) 2 + y 2 = a + ex for x > 0, –a – ex for x < 0, r 2 = (x – c) 2 + y 2 = –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 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 M 0 (x 0 , y 0 ) is described by the equation x 0 x a 2 – y 0 y b 2 = 1.(4.4.3.9) The distances d 1 and d 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 e – a| = r 2 Na , N = x 0 a 2 2 + y 0 b 2 2 ,(4.4.3.10) where r 1 and r 2 are the lengths of the focal radii of the point M 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 Y Figure 4.23. The tangent to the hyperbola (a). Optical property of a hyperbola (b). The tangent at any point M 0 (x 0 , y 0 ) of the hyperbola forms acute angles ϕ 1 and ϕ 2 with the focal radii of the point of tangency (see Fig. 4.23a), and sin ϕ 1 = d 1 r 1 = 1 Na ,sinϕ 2 = d 2 r 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 .(4.4.3.13) A hyperbola meets the diameter y = kx if and only if k 2 < b 2 a 2 .(4.4.3.14) The lengths l 1 and l 2 of the conjugate diameters with slopes k 1 and k 2 satisfy the relation l 1 l 2 sin(arctan k 2 –arctank 1 )=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 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 N Figure 4.24. Parabola. 4.4. SECOND-ORDER CURVES 105 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/2 is 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 √ p .(4.4.4.2) The area of the figure bounded by the parabola and the chord passing through the points M (x 1 , y 1 )andN(x 1 ,–y 1 ) is equal to (see Fig. 4.24b) S = 4 3 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: 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 M 0 (x 0 , y 0 ) is described by the equation yy 0 = p(x + x 0 ). (4.4.4.7) The direction vector of the tangent (4.4.4.7) has the coordinates (y 0 , p), and the direction vector of the line passing through the points M 0 (x 0 , y 0 )andF (p/2, 0) has the coordinates 106 ANALYTIC GEOMETRY O X 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 2 (x 0 – p/2) 2 + y 2 0 = y 0 y 2 0 + p 2 .(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 y 2 = 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 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 1 –cosϕ ,(4.4.4.12) where – 1 2 π ≤ ϕ ≤ 1 2 π. 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 a 11 x 2 + 2a 12 xy + a 22 y 2 + 2a 13 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) 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 a 2 + y 2 b 2 = 1, an ellipse; 2. x 2 a 2 – y 2 b 2 = 1, a hyperbola; 3. y = 2px, a parabola; 4. x 2 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; 7. x 2 – a 2 = 0, a pair of parallel straight lines; 8. x 2 + a 2 = 0, a pair of imaginary parallel straight lines; 9. x 2 = 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 = a 11 + a 22 , δ = a 22 a 23 a 32 a 33 , Δ = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ,(4.4.5.2) . is the locus of points for which the ratio of distances to the focus and the directrix is equal to 1: r |x + p/2| = 1.(4.4.4.6) 4.4.4-4. Equation of tangent and optical property of parabola. The. 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. (b). The tangent at any point M 0 (x 0 , y 0 ) of the hyperbola forms acute angles ϕ 1 and ϕ 2 with the focal radii of the point of tangency (see Fig. 4.23a), and sin ϕ 1 = d 1 r 1 = 1 Na ,sinϕ 2 = d 2 r 2 = 1 Na .(4.4.3.11) This