64 CHAPTER 8 Analytic Geometry 8.13. Find an equation for the perpendicular bisector of the line segment joining A(7,Ϫ8) and B(Ϫ2, 5). The perpendicular bisector of a segment consists of all points that are equidistant from its endpoints. Thus if PA ϭ PB, then P lies on the perpendicular bisector of AB. Let P have the unknown coordinates (x,y). Then, from the distance formula, PA ϭ PB if PA ϭ 2(x Ϫ 7)2 ϩ [y Ϫ (Ϫ8)]2 ϭ 2[x Ϫ (Ϫ2)]2 ϩ (y Ϫ 5)2 ϭ PB Squaring both sides and simplifying yields (x Ϫ 7)2 ϩ [y Ϫ (Ϫ8)]2 ϭ [x Ϫ (Ϫ2)]2 ϩ (y Ϫ 5)2 x2 Ϫ 14x ϩ 49 ϩ y2 ϩ 16y ϩ 64 ϭ x2 ϩ 4x ϩ 4 ϩ y2 Ϫ 10y ϩ 25 18x Ϫ 26y ϭ 84 9x Ϫ 13y ϭ 42 This is the equation satisfied by all points equidistant from A and B. Hence, it is the equation of the perpendicular bisector of AB. SUPPLEMENTARY PROBLEMS 8.14. Describe the set of points that satisfy the relations: (a) x ϭ 0; (b) x > 0; (c) xy < 0; (d) y > 1. Ans. (a) All points on the y-axis; (b) all points to the right of the y-axis; (c) all points in the second and fourth quadrants; (d) all points above the line y ϭ 1. 8.15. Find the distance between the following pairs of points: (a) (0,Ϫ7) and (7,0); (b) (Ϫ3 23,Ϫ3) and (3 23, 3), Ans. (a)7 22; (b) 12 8.16. Find the length and the midpoint of the line segment with the given endpoints: (a) A(1,8), B(Ϫ3,4); (b) A(3,Ϫ7), B(0,8); (c) A(1, 22), B(Ϫ1,5 22) 3 1 Ans.(a)length 4 22, midpoint (Ϫ1,6); (b)length 3 226, midpoint 2, 2 ; (c)length 6, midpoint (0,3 22) A B 8.17. Analyze the following for symmetry. Do not sketch graphs: (a) xy2 ϭ 4; (b) x3y ϭ 4; (c) ZxyZ ϭ 4; (d) x2 ϩ xy ϭ 4; (e) x2 ϩ y ϩ y2 ϭ 4; (f) x2 ϩ xy ϩ y2 ϭ 4 Ans. (a) x-axis symmetry; (b) origin symmetry; (c) x-axis, y-axis, origin symmetry; (d) origin symmetry; (e) y-axis symmetry; (f) origin symmetry 8.18. Analyze symmetry and intercepts, then sketch graphs of the following: (a) 3x ϩ 4y ϩ 12 ϭ 0(b)y2 ϭ 10 ϩ x (c) y2 Ϫ x2 ϭ 9(d) |y| Ϫ |x| ϭ 3 Ans. (a) Fig. 8-16: x-intercept Ϫ4, y-intercept Ϫ3, no symmetry Figure 8-16 CHAPTER 8 Analytic Geometry 65 (b) Fig. 8-17: x-intercept Ϫ10, y-intercepts Ϯ 210, x-axis symmetry Figure 8-17 (c) Fig. 8-18: no x-intercept, y-intercept Ϯ3, x-axis, y-axis, origin symmetry Figure 8-18 (d) Fig. 8-19: no x-intercept, y-intercepts Ϯ3, x-axis, y-axis, origin symmetry Figure 8-19 8.19. Analyze symmetry and intercepts, then sketch graphs of the following: y (a)x ϩ y ϭ 0 ; (b) y ϩ ZxZ ϭ 4; (c)x2 ϭ 4ZyZ ; 4 (d) ZyZ ϩ x2 ϭ 4; (e)ZxZ ϭ 4y2 ; (f) Ϫxy2 ϭ 4 2 Ans. (a) Fig. 8-20: x-intercepts 0, y-intercept 0, origin symmetry x –4 –2 2 4 –2 –4 Figure 8-20 66 CHAPTER 8 Analytic Geometry (b) Fig. 8-21: x-intercepts Ϯ4, y-intercept 4, y y-axis symmetry 4 3 2 1 x –6 –4 –2 2 4 6 –1 –2 Figure 8-21 (c) Fig. 8-22: x-intercept 0, y-intercept 0, x-axis, y y-axis, origin symmetry 2 1 x –6 –4 –2 2 4 6 –1 –2 Figure 8-22 (d) Fig. 8-23: x-intercepts Ϯ2, y-intercepts Ϯ4, x-axis, y y-axis, origin symmetry 15 10 5 x –6 –4 –2 2 4 6 –5 –10 –15 Figure 8-23 (e) Fig. 8-24: x-intercept 0, y-intercept 0, x-axis, y y-axis, origin symmetry 1.5 1 0.5 x –10 –5 5 10 –0.5 –1 –1.5 Figure 8-24 CHAPTER 8 Analytic Geometry 67 (f) Fig. 8-25: no intercepts, x-axis symmetry y 4 3 2 1 x –4 –3 –2 –1 1 2 3 4 –1 –2 –3 –4 Figure 8-25 1 5 4 8.20. Find the equations of the following circles: (a) center (5,Ϫ2), radius 210; (b) center 2,Ϫ2 , diameter 3; (c) center (3,8), passing through the origin; (d) center (Ϫ3,Ϫ4), tangent to the y-axis. A B 2 2 1 2 5 2 9 Ans. (a) (x Ϫ 5) ϩ (y ϩ 2) ϭ 210; (b) x Ϫ 2 ϩ y ϩ 2 ϭ 4; 2 2 2 2 (c) (x Ϫ 3) ϩ (y Ϫ 8) ϭ 73; (d) (x ϩA 3) ϩB (y ϩA 4) ϭB9 8.21. Find the equations of the following circles: (a) center (5,2), (3,Ϫ1) is a point on the circle; (b)(5,Ϫ5) and (Ϫ3,Ϫ9) are end points of a diameter. Ans.(a)(x Ϫ 5)2 ϩ (y Ϫ 2)2 ϭ 13; (b) (x Ϫ 1)2 ϩ (y ϩ 7)2 ϭ 20 8.22. For the following equations, determine whether they represent circles, and if so, find the center and radius: (a) x2 ϩ y2 ϩ 8x ϩ 2y ϭ 5; (b) x2 ϩ y2 Ϫ 4x Ϫ 8y ϩ 20 ϭ 0; (c) 2x2 ϩ 2y2 Ϫ 6x ϩ 14y ϭ 3; (d) x2 ϩ y2 ϩ 12x ϩ 20y ϩ 200 ϭ 0 Ans. (a) circle; center (Ϫ4, Ϫ1), radius 222; (b) this is not a circle; the graph consists only of the point (2,4); 3 7 (c) circle, center (2,Ϫ2), radius 4; (d) this is not a circle; there are no points on the graph. 8.23. Show that the triangle with vertices (Ϫ10,7), (Ϫ6,Ϫ2), and (3,2) is isosceles. 8.24. Show that the triangle with vertices (4, 23), (5,0), and (6, 23) is equilateral. 8.25. Show that the triangle with vertices , (1,1)(6,9) , and (9,Ϫ4) is an isosceles right triangle. 8.26. Show that the quadrilateral with vertices (Ϫ3,Ϫ3), (5,Ϫ1), (7,7), and (Ϫ1,5) is a rhombus. 8.27. Show that the quadrilateral with vertices (7,2), (10,0), (8,Ϫ3), and (5,Ϫ1) is a square. 8.28. (a) Find the equation of the perpendicular bisector of the line segment with endpoints (Ϫ2,Ϫ5) and (7,Ϫ1). (b) Show that the equation of the perpendicular bisector of the line segment with endpoints (x1,y1) and (x2,y2) x Ϫ x y Ϫ y can be written ϩ ϭ 0, where (x,y) are the coordinates of the midpoint of the segment. y2 Ϫ y1 x2 Ϫ x1 Ans. (a) 18x ϩ 8y ϭ 21.