ࡱ>  FHEuYJ}5@ /bjbj22 ,xXX'6666!!!!!< 82"4f(f(f((***7777777$8RP;7*[*@***766f((7555*6Rf(L(75*7555"5(&" 0x92!k+85W7470 85;3;56666;5**5*****77D u5CHAPTER 9 Algebra 2 Selected Answers Evens p.1 Section 9.1 (p.404) Oral Ex/ 1. 6 2. 6 3. 6 4. 6 5. 5 6. 10 7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9.  EMBED Equation.DSMT4  10. (3, 1) 11. (4,  EMBED Equation.DSMT4 ) 12.  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14. (4.5, 1.75) 15. (1.5, 3) Written Exercises  EMBED Equation.DSMT4   EMBED Equation.DSMT4  20. (4, -3) 22. (-6, 12) 24. (2h, 2k) 26. no, yes, 45 28. yes, yes, 9 30. collinear 32. collinear Problems 2. 2x y + 2 = 0 4. 5x 2y + 10 = 0 6. a 8. Midpoints are (0,5), (4,4), (3,-1), (-1,0). The slopes of opposite sides are equal -1/4 and 5. Therefore the quadrilateral is a parallelogram. 10. sample: Construct a line through one midpoint parallel to the line through the other two midpoints. Repeat two more times to get the triangle. 12. The square of the diagonals: (AC)2 = (a+b)2 + c2 and (BD)2 = (ab)2 + c2. Sides squared: (AB)2 = a2 = (CD)2 and (BC)2 = b2 + c2 = (AD)2. The sum of the squares of the 4 sides of  EMBED Equation.DSMT4 ABCD = 2a2 + 2b2 + 2c2. 14. 2 lines are perpendicular if and only if the product of their slopes is -1 (m1m2 = -1). See figure on page 406. The plan is to move 1 unit to the right of P(r,s) to get T1 & T2. Part (c) Prove the if part: m1m2 = -1  EMBED Equation.DSMT4  Notice T1(r + 1, s + m1) and T2(r + 1, s + m2) and P(r, s). (This is not that obvious!) Well then use the converse of the Pythagorean Theorem by showing that a2 + b2 = c2. (12 +  EMBED Equation.DSMT4 ) + (1 +  EMBED Equation.DSMT4 ) = (m2 m1)2 Is this true? Lets see: 2 +  EMBED Equation.DSMT4  +  EMBED Equation.DSMT4  =  EMBED Equation.DSMT4 2m2m1 + EMBED Equation.DSMT4  Remember we were given m1m2 = -1, so the 2s cancel out and yes, this is true. Therefore the two lines form a right angle. Part (d) Prove the only if part:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  m1m2 = -1 Going the other way, we have (12 +  EMBED Equation.DSMT4 ) + (1 +  EMBED Equation.DSMT4 ) = (m2 m1)2 as true from the Pythagorean Theorem (but we dont have m1m2 = -1). Expanding and canceling we get: 2 = -2m1m2 and hence m1m2 = -1 (which was to be demonstrated) Part (c) and Part (d) together gives us the mutual implication (iff):  EMBED Equation.DSMT4  Section 9.2 (p.409) Oral Ex/ 1. (0,0), R=7 2. (0,0), R=1 3. (2,4), R=6 4. (5, 0), R=5 5. (0,-5), R=1/3 6. (-4,3), R= EMBED Equation.DSMT4  7. x2 + y2 = 16 8. x2 + y2 = 2 9. (x 1)2 + (y 1)2 = 9 10. (x 3)2 + (y 2)2 = 25 11. (x 4)2 + y2 = 1 12. (x + 1)2 + y2 = 16 13. (x + 3)2 + (y + 3)2 = 9 14. (x + 3)2 + (y 4)2 = 8 15. (x 1)2 + (y + 1)2 = Written Exercises 2. x2 + (y + 1)2 = 1 4. (x + 3)2 + (y 1)2 = 25 6. (x + 4)2 + (y + 2)2 = 100 8. (x + 5)2 + (y 3)2 = 1/36 CHAPTER 9 Selected Answers Evens p.2 Section 8.2 (continued) - Written Exercises    16. (0,0), R=9 18. (3,0), R=3 20. (-5,2), R=3 22. (-6,3), R= EMBED Equation.DSMT4  24. (0,5/2) R=3/2    32. (-1/3, -1), R=1/3 34. (1,-4), R=3 36. (x + 2)2 + y2 = 16 38. (x + 3)2 + (y 4)2 = 9 40. (x 2)2 + (y 2)2 = 4 42. (x + 4)2 + (y 4)2 = 16 or (x + 4)2 + (y + 4)2 = 16   48.  EMBED Equation.DSMT4  50. Let P(x,y) be on the semicircle with A(-r,0) and B(r,0). Show m1m2 = -1 (hence perpendicular lines & rt angles).  EMBED Equation.DSMT4 . Their product is:  EMBED Equation.DSMT4  and since x2 + y2 = r2, we get: 1 Section 9.3 Oral Ex/ 1. D: x = 3 2. F(2,0) 3. V(2,6) 4. V(4,-5/2) 5. F(3,0) 6. D: y = -3 7. (0,0), |p| = 2, up 8. (0,0), |p| = 3, left 9. (0,-2), |p| = , left 10. (-5,0) |p| = , up 11. (-3,1), |p| = 1, right 12. (4,-4), |p| = 9, down CHAPTER 9 Selected Answers Evens p.3 Section 9.3 (continued) Written Exercises 2. F(2,6) 4. D: x = 5 6. V(0,0) [Note: The x and y scaling is different.] 8. y2 = -8(x 2) 10. y2 = 12x  12. x2 = 16y 14. (x + 2)2 = -6(y 3/2)  16. (y 1)2 = 4(x + 3)  (Ah Winplot vs TI-83 screen shots!) 18. V(0,0), F(-3/2,0), D: x = 3/2, Axis of Symmetry: y = 0 20. V(1,-1), F(1,-3/4), D: y = -5/4, Axis of Symmetry: x = 1 22. V(2,-3), F(0,-3), D: x = 4, Axis of Symmetry: y = -3 24. V(3,1), F(3,-3/2), D: y = 7/2, Axis of Symmetry: x = 3 26. V(4,1), F(13/4, 1), D: x = 19/4, Axis of Symmetry: y = 1     36. We use p (vs c) for the distance from vertex to focus (and to directrix). So (x,y) must be equidistant from F and directrix.  EMBED Equation.DSMT4  Squaring both sides:  EMBED Equation.DSMT4   EMBED Equation.DSMT4 or (x-h)2 = 4p(y-k) CHAPTER 9 Selected Answers Evens p.4 38. The equation of the line (chord)? with points (-a,a2) and (b,b2) and a slope of:  EMBED Equation.DSMT4  y b2 = (b a)(x b) point-slope form! Now at the y-intercept, x = 0. So plug a 0 in for x and solve for y (y-intercept). y = -b2 + ab + b2 = ab Q.E.D. Section 9.4 (p.421) Oral Ex/ 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  Written Exercises  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4         16.  EMBED Equation.DSMT4  18.  EMBED Equation.DSMT4  20.  EMBED Equation.DSMT4  22.  EMBED Equation.DSMT4  24. For  EMBED Equation.DSMT4  put a compass at the point the point (0,b) and make the radius = a. Foci are the x-intercepts. 26.  EMBED Equation.DSMT4  CHAPTER 9 Selected Answers Evens p.5    34. From the point (0,b) to the focal point (c,0) the distance must be a since the sum of the two distances to the two foci is always 2a and by symmetry, this distance is half of 2a. Then the Pythagorean Theorem gives: b2 + c2 = a2. Section 9.5 (p.430) Oral Ex/  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  7. The x-axis and the y-axis 8. Inverse Variation (rotated hyperbola) 9.  EMBED Equation.DSMT4  Written Exercises 2.  EMBED Equation.DSMT4   EMBED Equation.DSMT4        14.  EMBED Equation.DSMT4  16.  EMBED Equation.DSMT4  18.  EMBED Equation.DSMT4  CHAPTER 9 Selected Answers Evens p.6 Section 9.5 (continued)      30.  EMBED Equation.DSMT4  32.  EMBED Equation.DSMT4  34.  EMBED Equation.DSMT4  Section 9.6 (p.433) Oral Ex/ 1. ellipse, (-3,5), F1(-5,5), F2(-1,5) 2. hyperbola, (-7,-1), F1(-12,-1), F2(-2,-1) 3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6. (x + 1)2 + (y 1)2 = 9 7.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  9. e 10. a 11. d 12. b 13. f 14. c Written Exercises 2.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  8.  EMBED Equation.DSMT4  10.  EMBED Equation.DSMT4  12.  EMBED Equation.DSMT4  14. ellipse, (-1,1),  EMBED Equation.DSMT4  16. hyperbola, (1,-3),  EMBED Equation.DSMT4  18. hyperbola, (-2,1), F1(-2,-4), F2(-2,6) 14.  16.  18.  20. xy = -1/2 CHAPTER 9 Selected Answers Evens p.7 Section 9.7 (p.437) Oral Ex/ 1. circle & line, 2 2. circle & line, 0 3. circle & line, 1 4. parabola & line, 2 5. circle & parabola, 2 6. hyperbola & circle, 4 7. hyperbola & line, 1 8. circle & ellipse, 0 9. 10. yes, the axis of symmetry 11. yes, a line parallel to one of the asymptotes 12. no Written Exercises 2. two 4. zero 6. four 8. two     10. (#4) empty set 12. (#6)  EMBED Equation.DSMT4 &  EMBED Equation.DSMT4  14. (#8)  EMBED Equation.DSMT4     20a. Area: xy = 1, perimeter:  EMBED Equation.DSMT4  b. y2 = 4x + 4 c. (see above) Since the graphs intersect at only negative x & y values (it looks close though!), no such triangle. Section 9.8 (p.441) Oral Ex/ 1. (1, -5) 2. (-4, 2) 3. (3, 2) & (3, -2) CHAPTER 9 Selected Answers Evens p.8 Written Exercises 2. (40,7), (0,-3) 4. (2,2), (2,-2) 6. (-39/5, -2/5), (5,6) 8. (3,-2), (2,-3) 10. (-1,4) 12. no real solution 14.  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  16. no real solution 18.  EMBED Equation.DSMT4  20. (-2,3) 22.  EMBED Equation.DSMT4  24.  EMBED Equation.DSMT4  26.  EMBED Equation.DSMT4  Problems 2. 2 and , -2 and -1/2 4.  EMBED Equation.DSMT4  6. 7 m and 24 m 8.  EMBED Equation.DSMT4  10. 20 cm by 28 cm 12. 12 m 14.  EMBED Equation.DSMT4  & 262.4 m Section 9.9 (p.447) Oral Ex/ 1. (3,0,-4) 2. (0,2,2) 3. none 4. (-2,0,2) 5. (7,0,0), (2,3,2) 6. (0,0,-5), (-1,-2,-2) 7. (0,2,4), (0,4,2) 8. (0,0,0), (1,0,-3) 9. Add, 4 10. Subtract twice the first equation from the second, 1 11. Subtract twice the first equation from three times the second, 4 12. Add twice the first equation to 7 times the second, 19 Written Exercises 2. (3,1,-2) 4. (3,1,-2) 6. (-1,4,2) 8. (-2,1,-1) 10. (-3,1,-2) 12. (3,-3,4) 14. (5,-5,3) 16. (4,-2,2) 18. (-1,3,-2) 20a. (-3 2z, 3 + 2z, z) b. examples: (-5,5,1),(-3,3,0),(-9,9,3) 22. Elimination of variables results in the variables disappearing and an equation like 0 = 3. An inconsistent system. 24. (0,0,0) is the only solution Problems 2. 2 carats, 4 carats, 8 carats 4. 12g, 18g, 21g 6. 50cm by 10cm by 15cm 8. 678 10. y = x2 + 4x 2 12. x2 + y2 + 4x 21 = 0 Chapter 9 Review 2. b 4. b 6. d 8. b 10. c 12. b 14. a 16. c 18. c 20. b Chapter 9 Test 1a.  EMBED Equation.DSMT4  b. (-1,-1) c. (11,-14) 2. (x 6)2 + (y + 3)2 = 116 3. C(4,-1), R =  EMBED Equation.DSMT4     11. 3  12. 2  10. hyperbola, (2,-1)  EMBED Equation.DSMT4  13.  EMBED Equation.DSMT4  14.  EMBED Equation.DSMT4  15. (-1,5,-2) 5. V(1,-3), F(3,-3), D: x = -1 6.  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