Identifying second degree equations


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1 Chapter 7 Identifing second degree equations 7.1 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, (7.1) 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 6.1, equation 6.2 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, 1 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 7.1 takes the form a 2 + b 2 + 2g + 2f + c = 0. (7.2) Then b completing the square and suitabl translating the 1, 1 aes, to new 2, 2 aes, equation 7.2 can be reduced to one of several standard forms, each of which is eas to sketch. We need some preliminar definitions. 129
2 0 CHAPTER 7. IDENTIFYING SECOND DEGREE EQUATIONS DEFINITION (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 detp = ±1. For det (P t P) = detp t det P = ( detp) 2, so (det P) 2 = det I n = 1. Hence det P = ±1. If P is an orthogonal matri with detp = 1, then P is called a proper orthogonal matri. THEOREM If P is a 2 2 orthogonal matri with detp = 1, then cos θ sinθ P = sinθ cos θ for some θ. REMARK 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 gives Hence a = d, b = c and so 1 P t = P 1 = 1 adjp a c b d = a c P = c a d b c a where a 2 + c 2 = 1. But then the point (a, c) lies on the unit circle, so a = cos θ and c = sinθ, where θ is uniquel determined up to multiples of 2π.,
3 7.1. THE EIGENVALUE METHOD 1 a DEFINITION (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 (Geometrical meaning 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. (7.3) Now AB 2 = ( 2 1 ) 2 + ( 2 1 ) 2, OA 2 = , OB2 = Substituting in equation 7.3 then gives ( 2 1 ) 2 + ( 2 1 ) 2 = ( ) + ( ) 2OA OB cos θ,
4 2 CHAPTER 7. IDENTIFYING SECOND DEGREE EQUATIONS which simplifies to give OA OB cos θ = = X Y. It follows from theorem 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 (Orthogonal vectors) Vectors X and Y are called orthogonal if X Y = 0. There is also a connection with orthogonal matrices: THEOREM 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 = 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 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
5 7.1. THE EIGENVALUE METHOD 3 Proof. Suppose AX 1 = λ 1 X 1, AX 2 = λ 2 X 2, (7.4) 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 7.4, X t 2AX 1 = λ 1 X t 2X 1 (7.) and X t 1AX 2 = λ 2 X t 1X 2. (7.6) 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. (7.7) Finall, subtracting equation 7.6 from equation 7.7, we have (λ 1 λ 2 )X t 1X 2 = 0 and hence, since λ 1 λ 2, X t 1X 2 = 0. THEOREM 7.1. 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 λ
6 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 7.1.4, X 1 and X 2 are orthogonal. B dividing X 1 and X 2 b their lengths (i.e. normalizing X 1 and X 2 ) if necessar, we can assume that X 1 and X 2 have unit length. Then b theorem 7.1.1, P = X 1 X 2 is an orthogonal matri. B replacing X 1 b X 1, if necessar, we can assume that detp = 1. Then b theorem 6.2.1, we have Also under the rotation X = PY, P t AP = P 1 λ1 0 AP = 0 λ 2 X t AX = (PY ) t A(PY ) = Y t (P t AP)Y = Y t diag (λ 1, λ 2 )Y = λ λ EXAMPLE Let A be the smmetric matri A = 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 = and X 2 = Then if P = X 1 X 2, the proof of theorem 7.1. shows that P t AP = However det P = 1, so replacing X 1 b X 1 will give det P = 1..
7 7.1. THE EIGENVALUE METHOD Figure 7.1: = 0. REMARK (A shortcut) Once we have determined one eigenvector X 1 =, the other can be taken to be, as these vectors are a b b a alwas orthogonal. Also P = X 1 X 2 will have detp = a 2 + b 2 > 0. We now appl the above ideas to determine the geometric nature of second degree equations in and. EXAMPLE Sketch the curve determined b the equation = 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 predicts, P is a rotation matri and the transformation 1 X = = PY = P 1,
8 6 CHAPTER 7. IDENTIFYING SECOND DEGREE EQUATIONS or more eplicitl = , = , (7.8) 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 , = Then under the rotation X = PY, our original quadratic equation becomes ( ) 38 ( ) + 31 = 0, or = 0. Now complete the square in 1 and 1 : ( ) ( ) = 0, ( ) 2 ( ) 2 ( ) 8 2 ( ) 1 2 = = 48. (7.9) Then if we perform a translation of aes to the new origin ( 1, 1 ) = ( 8, 1 ): 2 = 1 + 8, 2 = 1 + 1, equation 7.9 reduces to or = 48, = 1.
9 7.1. THE EIGENVALUE METHOD 7 Figure 7.2: 2 a = 1, 0 < b < a: an ellipse. b2 This equation is now in one of the standard forms listed below as Figure 7.2 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 7.8 can be solved to give 1 = , 1 =. Hence the 2 ais is given b 0 = 2 = = , or = 0. Similarl the 2 ais is given b = 0. This ellipse is sketched in Figure 7.1. Figures 7.2, 7.3, 7.4 and 7. are a collection of standard second degree equations: Figure 7.2 is an ellipse; Figures 7.3 are hperbolas (in both these eamples, the asmptotes are the lines = ± b ); Figures 7.4 and 7. a represent parabolas. EXAMPLE Sketch = 0.
10 8 CHAPTER 7. IDENTIFYING SECOND DEGREE EQUATIONS Figure 7.3: (i) 2 a 2 2 b 2 = 1; 2 (ii) a 2 2 = 1, 0 < b, 0 < a. b2 Figure 7.4: (i) 2 = 4a, a > 0; (ii) 2 = 4a, a < 0.
11 7.1. THE EIGENVALUE METHOD 9 Figure 7.: (iii) 2 = 4a, a > 0; (iv) 2 = 4a, a < 0. Solution. Complete the square: = 0 ( ) 2 = = 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), i.e. (, ) = ( 8, ). The parabola is sketched in Figure 7.6. EXAMPLE Sketch the curve = 0. Solution. We have = 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 = Then P = X 1 X 2 is a proper orthogonal matri and under the rotation of aes X = PY, or = = ,
12 140 CHAPTER 7. IDENTIFYING SECOND DEGREE EQUATIONS Figure 7.6: = 0. we have = λ λ = 2 1. The original quadratic equation becomes ( ) 9 = 0 ( ) = 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), i.e. ( 1, 1 ) = ( 1, 2 ), or (, ) = ( 21, 8 ). The ais of smmetr of the parabola is the line 2 = 0, i.e. 1 = 1/. Using the rotation equations in the form 1 = 2
13 7.2. A CLASSIFICATION ALGORITHM Figure 7.7: = 0. 1 = 2 +, we have 2 = 1, or 2 = 1. The parabola is sketched in Figure A classification algorithm There are several possible degenerate cases that can arise from the general second degree equation. For eample = 0 represents the point (0, 0); = 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.
14 142 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. (7.10) 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 h f b C, β = a g h f C. (7.11) (1.1) C 0. Translate aes to the new origin (α, β), where α and β are given b equations 7.11: Then equation 7.10 reduces to (1.2) C = 0. = 1 + α, = 1 + β. a h 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.
15 7.2. A CLASSIFICATION ALGORITHM 143 (C) A < 0: Two parallel lines (ii) b = f = 0. (b) h 0. CASE (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. (2.1) C 0. Translate aes to the new origin (α, β), where α and β are given b equations 7.11: = 1 + α, = 1 + β. Equation 7.10 becomes a h b 2 1 = C. (7.12) CASE 2.1(i) h = 0. Equation 7.12 becomes a 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.
16 144 CHAPTER 7. IDENTIFYING SECOND DEGREE EQUATIONS CASE 2.1(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 7.12 becomes where Here λ 1 λ 2 = C. λ λ = 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 v 2 =, where u =, v =. Cλ 1 Cλ 2 (2.1) 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 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
17 7.2. 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 + hf a + b. The verte of the parabola is ( (2akf hk 2 hac) d ), a(k2 + ac 2kg), d where d = 2a(gh af). 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 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 Identif the curve = 0. (7.14)
18 146 CHAPTER 7. IDENTIFYING SECOND DEGREE EQUATIONS Solution. Here = = 0. Let = 1 + α, = 1 + β and substitute in equation 7.14 to get 2( 1 + α) 2 + ( 1 + α)( 1 + β) ( 1 + β) 2 + 4( 1 + β) 8 = 0. (7.1) 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 7.1 simplifies to = 0 = (2 1 1 )( ), so relative to the 1, 1 coordinates, equation 7.14 describes two lines: = 0 or = 0. In terms of the original, coordinates, these lines become 2( ) ( 8 3 ) = 0 and ( ) + ( 8 3 ) = 0, i.e = 0 and + 2 = 0, which intersect in the point (, ) = (α, β) = ( 2 3, 8 3 ). EXAMPLE Identif the curve Solution. Here = 0. (7.16) = = 0. Let = 1 + α, = 1 + β and substitute in equation 7.16 to get ( 1 +α) 2 +2( 1 +α)( 1 +β)+( 1 +β) 2 +2( 1 +α)+2( 1 +β)+1 = 0. (7.17) 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 7.17 simplifies to = 0 = ( ) 2, and in terms of, coordinates, equation 7.16 becomes ( + + 1) 2 = 0, or = 0.
19 7.3. PROBLEMS PROBLEMS 1. Sketch the curves (i) = 0; (ii) = Sketch the hperbola = 8 and find the equations of the asmptotes. Answer: = 0 and = Sketch the ellipse = 36 and find the equations of the aes of smmetr. Answer: = 2 and = 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) = 0; (ii) = 0; (iii) = 0; (iv) = 0. Answers: (i) hperbola, centre (3, 2), asmptotes = 0, = 0; (ii) ellipse, centre (0, ); (iii) parabola, verte ( 7, 9 ), ais of smmetr = 0; (iv) hperbola, centre ( 1 10, 7 10 ), asmptotes = 0 and = 0.. Identif the lines determined b the equations: (i) = 0;
20 148 CHAPTER 7. IDENTIFYING SECOND DEGREE EQUATIONS (ii) = 0; (iii) = 0. Answers: (i) = 0 and + 1 = 0; (ii) = 0; (iii) = 0 and = 0.
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