Identifying second degree equations

Chapter 7 Identifing second degree equations 71 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g + 2f + c = 0, (71) where not all of a, h, b are zero a h Let A = be the matri of the quadratic form a h b 2 + 2h + b 2 We saw in section 61, equation 62 that A has real eigenvalues λ 1 and λ 2, given b λ 1 = a + b (a b) 2 + 4h 2 2, λ 2 = a + b + (a b) 2 + 4h 2 2 We show that it is alwas possible to rotate the, aes to 1, 2 aes whose positive directions are determined b eigenvectors X 1 and X 2 corresponding to λ 1 and λ 2 in such a wa that relative to the 1, 1 aes, equation 71 takes the form a 2 + b 2 + 2g + 2f + c = 0 (72) Then b completing the square and suitabl translating the 1, 1 aes, to new 2, 2 aes, equation 72 can be reduced to one of several standard forms, each of which is eas to sketch We need some preliminar definitions 129

0 CHAPTER 7 IDENTIFYING SECOND DEGREE EQUATIONS DEFINITION 711 (Orthogonal matri) An n n real matri P is called orthogonal if P t P = I n It follows that if P is orthogonal, then det P = ±1 For det (P t P ) = det P t det P = ( det P ) 2, so (det P ) 2 = det I n = 1 Hence det P = ±1 If P is an orthogonal matri with det P = 1, then P is called a proper orthogonal matri THEOREM 711 If P is a 2 2 orthogonal matri with det P = 1, then cos θ sin θ P = sin θ cos θ for some θ REMARK 711 Hence, b the discusssion at the beginning of Chapter 6, if P is a proper orthogonal matri, the coordinate transformation 1 = P represents a rotation of the aes, with new 1 and 1 aes given b the repective columns of P Proof Suppose that P t P = I 2, where = det P = 1 Let a b P = c d Then the equation 1 P t = P 1 = 1 adj P gives a c b d Hence a = d, b = c and so = d c a c P = c a b a, where a 2 + c 2 = 1 But then the point (a, c) lies on the unit circle, so a = cos θ and c = sin θ, where θ is uniquel determined up to multiples of 2π

71 THE EIGENVALUE METHOD 1 a DEFINITION 712 (Dot product) If X = b X Y, the dot product of X and Y, is defined b and Y = c d, then X Y = ac + bd The dot product has the following properties: (i) X (Y + Z) = X Y + X Z; (ii) X Y = Y X; (iii) (tx) Y = t(x Y ); (iv) X X = a 2 + b 2 if X = (v) X Y = X t Y The length of X is defined b a b ; X = a 2 + b 2 = (X X) 1/2 We see that X is the distance between the origin O = (0, 0) and the point (a, b) THEOREM 712 (Geometrical interpretation of the dot product) Let A = ( 1, 1 ) and B = ( 2, 2 ) be points, each distinct from the origin 1 2 O = (0, 0) Then if X = and Y =, we have 1 2 X Y = OA OB cos θ, where θ is the angle between the ras OA and OB Proof B the cosine law applied to triangle OAB, we have AB 2 = OA 2 + OB 2 2OA OB cos θ (73) Now AB 2 = ( 2 1 ) 2 + ( 2 1 ) 2, OA 2 = 2 1 + 2 1, OB2 = 2 2 + 2 2 Substituting in equation 73 then gives ( 2 1 ) 2 + ( 2 1 ) 2 = ( 2 1 + 2 1) + ( 2 2 + 2 2) 2OA OB cos θ,

2 CHAPTER 7 IDENTIFYING SECOND DEGREE EQUATIONS which simplifies to give OA OB cos θ = 1 2 + 1 2 = X Y It follows from theorem 712 that if A = ( 1, 1 ) and B = ( 2, 2 ) are 1 2 points distinct from O = (0, 0) and X = and Y =, then X Y = 0 means that the ras OA and OB are perpendicular This is the reason for the following definition: DEFINITION 7 (Orthogonal vectors) Vectors X and Y are called orthogonal if X Y = 0 There is also a connection with orthogonal matrices: THEOREM 7 Let P be a 2 2 real matri Then P is an orthogonal matri if and onl if the columns of P are orthogonal and have unit length Proof P is orthogonal if and onl if P t P = I 2 Now if P = X 1 X 2, the matri P t P is an important matri called the Gram matri of the column vectors X 1 and X 2 It is eas to prove that 1 P t X1 X P = X i X j = 1 X 1 X 2 X 2 X 1 X 2 X 2 Hence the equation P t P = I 2 is equivalent to X1 X 1 X 1 X 2 X 2 X 1 X 2 X 2 = 1 0 0 1 or, equating corresponding elements of both sides:, X 1 X 1 = 1, X 1 X 2 = 0, X 2 X 2 = 1, which sas that the columns of P are orthogonal and of unit length The net theorem describes a fundamental propert of real smmetric matrices and the proof generalizes to smmetric matrices of an size THEOREM 714 If X 1 and X 2 are eigenvectors corresponding to distinct eigenvalues λ 1 and λ 2 of a real smmetric matri A, then X 1 and X 2 are orthogonal vectors 2

71 THE EIGENVALUE METHOD 3 Proof Suppose AX 1 = λ 1 X 1, AX 2 = λ 2 X 2, (74) where X 1 and X 2 are non zero column vectors, A t = A and λ 1 λ 2 We have to prove that X t 1 X 2 = 0 From equation 74, X t 2AX 1 = λ 1 X t 2X 1 (7) and X t 1AX 2 = λ 2 X t 1X 2 (76) From equation 7, taking transposes, (X t 2AX 1 ) t = (λ 1 X t 2X 1 ) t so Hence X t 1A t X 2 = λ 1 X t 1X 2 X t 1AX 2 = λ 1 X t 1X 2 (77) Finall, subtracting equation 76 from equation 77, we have (λ 1 λ 2 )X t 1X 2 = 0 and hence, since λ 1 λ 2, X t 1X 2 = 0 THEOREM 71 Let A be a real 2 2 smmetric matri with distinct eigenvalues λ 1 and λ 2 Then a proper orthogonal 2 2 matri P eists such that P t AP = diag (λ 1, λ 2 ) Also the rotation of aes = P 1 1 diagonalizes the quadratic form corresponding to A: X t AX = λ 1 2 1 + λ 2 2 1

4 CHAPTER 7 IDENTIFYING SECOND DEGREE EQUATIONS Proof Let X 1 and X 2 be eigenvectors corresponding to λ 1 and λ 2 Then b theorem 714, X 1 and X 2 are orthogonal B dividing X 1 and X 2 b their lengths (ie normalizing X 1 and X 2 ) if necessar, we can assume that X 1 and X 2 have unit length Then b theorem 711, P = X 1 X 2 is an orthogonal matri B replacing X 1 b X 1, if necessar, we can assume that det P = 1 Then b theorem 621, we have Also under the rotation X = P Y, P t AP = P 1 λ1 0 AP = 0 λ 2 X t AX = (P Y ) t A(P Y ) = Y t (P t AP )Y = Y t diag (λ 1, λ 2 )Y = λ 1 2 1 + λ 2 2 1 EXAMPLE 711 Let A be the smmetric matri A = 12 6 6 7 Find a proper orthogonal matri P such that P t AP is diagonal Solution The characteristic equation of A is λ 2 19λ + 48 = 0, or (λ 16)(λ 3) = 0 Hence A has distinct eigenvalues λ 1 = 16 and λ 2 = 3 We find corresponding eigenvectors 3 2 X 1 = and X 2 2 = 3 Now X 1 = X 2 = So we take X 1 = 1 3 2 and X 2 = 1 2 3 Then if P = X 1 X 2, the proof of theorem 71 shows that P t AP = 16 0 0 3 However det P = 1, so replacing X 1 b X 1 will give det P = 1

71 THE EIGENVALUE METHOD 2 4 2-4 -2 2 4-2 2-4 Figure 71: 12 2 12 + 7 2 + 60 38 + 31 = 0 REMARK 712 (A shortcut) Once we have determined one eigenvector X 1 =, the other can be taken to be, as these these vectors a b b a are alwas orthogonal Also P = X 1 X 2 will have det P = a 2 + b 2 > 0 We now appl the above ideas to determine the geometric nature of second degree equations in and EXAMPLE 712 Sketch the curve determined b the equation 12 2 12 + 7 2 + 60 38 + 31 = 0 Solution With P taken to be the proper orthogonal matri defined in the previous eample b P = 3/ 2/ 2/ 3/ then as theorem 711 predicts, P is a rotation matri and the transformation 1 X = = P Y = P 1,

6 CHAPTER 7 IDENTIFYING SECOND DEGREE EQUATIONS or more eplicitl = 3 1 + 2 1, = 2 1 + 3 1, (78) will rotate the, aes to positions given b the respective columns of P (More generall, we can alwas arrange for the 1 ais to point either into the first or fourth quadrant) 12 6 Now A = is the matri of the quadratic form 6 7 so we have, b Theorem 71 12 2 12 + 7 2, 12 2 12 + 7 2 = 16 2 1 + 3 2 1 Then under the rotation X = P Y, our original quadratic equation becomes 16 2 1 + 3 2 1 + 60 (3 1 + 2 1 ) 38 ( 2 1 + 3 1 ) + 31 = 0, or 16 2 1 + 3 2 1 + 26 1 + 6 1 + 31 = 0 Now complete the square in 1 and 1 : ( 16 2 1 + 16 ) ( 1 + 3 1 2 + 2 ) 1 + 31 = 0, ( 16 1 + 8 ) 2 ( + 3 1 + 1 ) 2 ( ) 8 2 ( ) 1 2 = 16 + 3 31 = 48 (79) Then if we perform a translation of aes to the new origin ( 1, 1 ) = ( 8, 1 ): 2 = 1 + 8, 2 = 1 + 1, equation 79 reduces to or 16 2 2 + 3 2 2 = 48, 2 2 3 + 2 2 16 = 1

71 THE EIGENVALUE METHOD 7 Figure 72: 2 a 2 + 2 = 1, 0 < b < a: an ellipse b2 This equation is now in one of the standard forms listed below as Figure 72 and is that of a whose centre is at ( 2, 2 ) = (0, 0) and whose aes of smmetr lie along the 2, 2 aes In terms of the original, coordinates, we find that the centre is (, ) = ( 2, 1) Also Y = P t X, so equations 78 can be solved to give Hence the 2 ais is given b 1 = 3 1 2 1, 1 = 2 1 + 3 1 0 = 2 = 1 + 8 = 3 2 + 8, or 3 2 + 8 = 0 Similarl the 2 ais is given b 2 + 3 + 1 = 0 This ellipse is sketched in Figure 71 Figures 72, 73, 74 and 7 are a collection of standard second degree equations: Figure 72 is an ellipse; Figures 73 are hperbolas (in both these eamples, the asmptotes are the lines = ± b ); Figures 74 and 7 a represent parabolas EXAMPLE 7 Sketch 2 4 10 7 = 0

8 CHAPTER 7 IDENTIFYING SECOND DEGREE EQUATIONS Figure 73: (i) 2 a 2 2 b 2 = 1; 2 (ii) a 2 2 = 1, 0 < b, 0 < a b2 Figure 74: (i) 2 = 4a, a > 0; (ii) 2 = 4a, a < 0

71 THE EIGENVALUE METHOD 9 Figure 7: (iii) 2 = 4a, a > 0; (iv) 2 = 4a, a < 0 Solution Complete the square: 2 10 + 2 4 32 = 0 ( ) 2 = 4 + 32 = 4( + 8), or 2 1 = 4 1, under the translation of aes 1 = + 8, 1 = Hence we get a parabola with verte at the new origin ( 1, 1 ) = (0, 0), ie (, ) = ( 8, ) The parabola is sketched in Figure 76 EXAMPLE 714 Sketch the curve 2 4 + 4 2 + 9 = 0 Solution We have 2 4 + 4 2 = X t AX, where 1 2 A = 2 4 The characteristic equation of A is λ 2 λ = 0, so A has distinct eigenvalues λ 1 = and λ 2 = 0 We find corresponding unit length eigenvectors X 1 = 1 1, X 2 2 = 1 2 1 Then P = X 1 X 2 is a proper orthogonal matri and under the rotation of aes X = P Y, or = 1 + 2 1 = 2 1 + 1,

140 CHAPTER 7 IDENTIFYING SECOND DEGREE EQUATIONS 1 12 8 4 1-8 -4 4 8 12-4 -8 Figure 76: 2 4 10 7 = 0 we have 2 4 + 4 2 = λ 1 2 1 + λ 2 2 1 = 2 1 The original quadratic equation becomes 2 1 + ( 2 1 + 1 ) 9 = 0 ( 2 1 2 1 ) + 1 9 = 0 ( 1 1 ) 2 = 10 1 = ( 1 2 ), or 2 2 = 1 2, where the 1, 1 aes have been translated to 2, 2 aes using the transformation 2 = 1 1, 2 = 1 2 Hence the verte of the parabola is at ( 2, 2 ) = (0, 0), ie ( 1, 1 ) = ( 1, 2 ), or (, ) = ( 21, 8 ) The ais of smmetr of the parabola is the line 2 = 0, ie 1 = 1/ Using the rotation equations in the form 1 = 2

72 A CLASSIFICATION ALGORITHM 141 4 2 2-4 -2 2 4 2-2 -4 Figure 77: 2 4 + 4 2 + 9 = 0 1 = 2 +, we have 2 = 1, or 2 = 1 The parabola is sketched in Figure 77 72 A classification algorithm There are several possible degenerate cases that can arise from the general second degree equation For eample 2 + 2 = 0 represents the point (0, 0); 2 + 2 = 1 defines the empt set, as does 2 = 1 or 2 = 1; 2 = 0 defines the line = 0; ( + ) 2 = 0 defines the line + = 0; 2 2 = 0 defines the lines = 0, + = 0; 2 = 1 defines the parallel lines = ±1; ( + ) 2 = 1 likewise defines two parallel lines + = ±1 We state without proof a complete classification 1 of the various cases 1 This classification forms the basis of a computer program which was used to produce the diagrams in this chapter I am grateful to Peter Adams for his programming assistance

142 CHAPTER 7 IDENTIFYING SECOND DEGREE EQUATIONS that can possibl arise for the general second degree equation a 2 + 2h + b 2 + 2g + 2f + c = 0 (710) It turns out to be more convenient to first perform a suitable translation of aes, before rotating the aes Let a h g = h b f g f c, C = ab h2, A = bc f 2, B = ca g 2 If C 0, let CASE 1 = 0 α = g f C h b, β = a h C g f (711) (11) C 0 Translate aes to the new origin (α, β), where α and β are given b equations 711: Then equation 710 reduces to (12) C = 0 = 1 + α, = 1 + β a 2 1 + 2h 1 1 + b 2 1 = 0 (a) C > 0: Single point (, ) = (α, β) (b) C < 0: Two non parallel lines intersecting in (, ) = (α, β) The lines are β α = h ± C if b 0, b β = α and α = a, if b = 0 2h (a) h = 0 (i) a = g = 0 (A) A > 0: Empt set (B) A = 0: Single line = f/b

72 A CLASSIFICATION ALGORITHM 143 (C) A < 0: Two parallel lines (ii) b = f = 0 (b) h 0 CASE 2 0 (A) B > 0: Empt set = f ± A b (B) B = 0: Single line = g/a (C) B < 0: Two parallel lines (i) B > 0: Empt set = g ± B a (ii) B = 0: Single line a + h = g (iii) B < 0: Two parallel lines a + h = g ± B (21) C 0 Translate aes to the new origin (α, β), where α and β are given b equations 711: = 1 + α, = 1 + β Equation 710 becomes a 2 1 + 2h 1 1 + b 2 1 = C (712) CASE 21(i) h = 0 Equation 712 becomes a 2 1 + b2 1 = C (a) C < 0: Hperbola (b) C > 0 and a > 0: Empt set (c) C > 0 and a < 0 (i) a = b: Circle, centre (α, β), radius (ii) a b: Ellipse g 2 +f 2 ac a

144 CHAPTER 7 IDENTIFYING SECOND DEGREE EQUATIONS CASE 21(ii) h 0 Rotate the ( 1, 1 ) aes with the new positive 2 ais in the direction of (b a + R)/2, h, where R = (a b) 2 + 4h 2 Then equation 712 becomes where Here λ 1 λ 2 = C λ 1 2 2 + λ 2 2 2 = C (7) λ 1 = (a + b R)/2, λ 2 = (a + b + R)/2, (a) C < 0: Hperbola Here λ 2 > 0 > λ 1 and equation 7 becomes 2 2 u 2 2 2 v 2 =, where u =, v = Cλ 1 Cλ 2 (21) C = 0 (b) C > 0 and a > 0: Empt set (c) C > 0 and a < 0: Ellipse Here λ 1, λ 2, a, b have the same sign and λ 1 λ 2 and equation 7 becomes 2 2 u 2 + 2 2 v 2 = 1, where (a) h = 0 u =, v = Cλ 1 Cλ 2 (i) a = 0: Then b 0 and g 0 Parabola with verte ( ) A 2gb, f b

72 A CLASSIFICATION ALGORITHM 14 Translate aes to ( 1, 1 ) aes: 2 1 = 2g b 1 (ii) b = 0: Then a 0 and f 0 Parabola with verte ( g a, B ) 2fa Translate aes to ( 1, 1 ) aes: (b) h 0: Parabola Let 2 1 = 2f a 1 k = ga + bf a + b The verte of the parabola is ( (2akf hk 2 hac) d ), a(k2 + ac 2kg) d Now translate to the verte as the new origin, then rotate to ( 2, 2 ) aes with the positive 2 ais along sa, sh, where s = sign (a) (The positive 2 ais points into the first or fourth quadrant) Then the parabola has equation 2 2 = 2st a 2 + h 2 2, where t = (af gh)/(a + b) REMARK 721 If = 0, it is not necessar to rotate the aes Instead it is alwas possible to translate the aes suitabl so that the coefficients of the terms of the first degree vanish EXAMPLE 721 Identif the curve 2 2 + 2 + 6 8 = 0 (714)

146 CHAPTER 7 IDENTIFYING SECOND DEGREE EQUATIONS Solution Here = 1 2 2 0 1 2 1 3 0 3 8 = 0 Let = 1 + α, = 1 + β and substitute in equation 714 to get 2( 1 + α) 2 + ( 1 + α)( 1 + β) ( 1 + β) 2 + 4( 1 + β) 8 = 0 (71) Then equating the coefficients of 1 and 1 to 0 gives 4α + β = 0 α + 2β + 4 = 0, which has the unique solution α = 2 3, β = 8 3 Then equation 71 simplifies to 2 2 1 + 1 1 1 2 = 0 = (2 1 1 )( 1 + 1 ), so relative to the 1, 1 coordinates, equation 714 describes two lines: 2 1 1 = 0 or 1 + 1 = 0 In terms of the original, coordinates, these lines become 2( + 2 3 ) ( 8 3 ) = 0 and ( + 2 3 ) + ( 8 3 ) = 0, ie 2 + 4 = 0 and + 2 = 0, which intersect in the point (, ) = (α, β) = ( 2 3, 8 3 ) EXAMPLE 722 Identif the curve Solution Here 2 + 2 + 2 + +2 + 2 + 1 = 0 (716) = 1 1 1 1 1 1 1 1 1 = 0 Let = 1 + α, = 1 + β and substitute in equation 716 to get ( 1 +α) 2 +2( 1 +α)( 1 +β)+( 1 +β) 2 +2( 1 +α)+2( 1 +β)+1 = 0 (717) Then equating the coefficients of 1 and 1 to 0 gives the same equation 2α + 2β + 2 = 0 Take α = 0, β = 1 Then equation 717 simplifies to 2 1 + 2 1 1 + 2 1 = 0 = ( 1 + 1 ) 2, and in terms of, coordinates, equation 716 becomes ( + + 1) 2 = 0, or + + 1 = 0

73 PROBLEMS 147 73 PROBLEMS 1 Sketch the curves (i) 2 8 + 8 + 8 = 0; (ii) 2 12 + 2 + 2 = 0 2 Sketch the hperbola 4 3 2 = 8 and find the equations of the asmptotes Answer: = 0 and = 4 3 3 Sketch the ellipse 8 2 4 + 2 = 36 and find the equations of the aes of smmetr Answer: = 2 and = 2 4 Sketch the conics defined b the following equations Find the centre when the conic is an ellipse or hperbola, asmptotes if an hperbola, the verte and ais of smmetr if a parabola: (i) 4 2 9 2 24 36 36 = 0; (ii) 2 4 + 8 2 + 4 16 + 4 = 0; (iii) 4 2 + 2 4 10 19 = 0; (iv) 77 2 + 78 27 2 + 70 30 + 29 = 0 Answers: (i) hperbola, centre (3, 2), asmptotes 2 3 12 = 0, 2 + 3 = 0; (ii) ellipse, centre (0, ); (iii) parabola, verte ( 7, 9 ), ais of smmetr 2 + 1 = 0; (iv) hperbola, centre ( 1 10, 7 10 ), asmptotes 7 + 9 + 7 = 0 and 11 3 1 = 0 Identif the lines determined b the equations: (i) 2 2 + 2 + 3 4 + 3 = 0;

148 CHAPTER 7 IDENTIFYING SECOND DEGREE EQUATIONS (ii) 9 2 + 2 6 + 6 2 + 1 = 0; (iii) 2 + 4 + 4 2 2 2 = 0 Answers: (i) 2 + 3 = 0 and + 1 = 0; (ii) 3 + 1 = 0; (iii) + 2 + 1 = 0 and + 2 2 = 0