1. Math
  2. Advanced Math
  3. complete 1724 32 34 38...

Question: complete 1724 32 34 38...

Question details

complete 17,24, 32, 34, 38

In Exercises 11 to 17, write the equation of the given circle or straight line in complex number notation. For example, the circle of radius 4 centered at the point 3- 2i is given by the equation z- (3-2i)-4. 11·The circle of radius 2 centered at 4 + i. 12. The straight line through 1 and -1-i. 13. The vertical line containing -3-i. 14. The circle through 0, 2 2i, and 2 - 2i. 15. The circle through 1, i, and 0. 16. The perpendicular bisector of the line segment joining-1 2i and 1 - 2i. 17. The straight line of slope -2 through 1 - i. 18. Show that the two lines Re(az b)-0 and Re(cz d)-0 are perpendicular if and only if Re(ac)-0 19. Let p be a positive real number and let Г be the locus of points z satisfying Iz-pl-cx, z-x + iy. Show that Γ is (a) an ellipse if 0 < c < 1; (b) a parabola if c = 1; (c) a hyperbola if 1 < c < ao 20. Let z, and z2 be distinct complex numbers. Show that the locus of points tzı + (1-t)22,-oo < t < ao, describes the line through z, and Z2. The values 0S< give the line segment joining z, and zz 21. Let α be a complex number with 0 < l < 1. Show that the set of all z with (a) Iz-α! <\ 1-azl is the disc {z: Izl < 1} (b) Iz _ α1 = 11-azl is the circle {z: Izl-1) (c) I-α/> I 1-azl is the set {z: 비 > 1} (Hint: Square both sides and simplify.) 22. Let z and w be nonzero complex numbers. Show that z wzw if and only if z sw for some positive real number s. In Exercises 23 to 26, follow the technique outlined in the text to find all solutions of the given equation. 27. Suppose that n is an odd integer and w is a negative real number. Show that 28. Let a, b, and c be complex numbers with a 0. Show that the solutions of one solution of the equation -w is a negative real number. (For instance, -2 is a root of z38.) az2 + bz cOare 21,22-bt b1 - 4ac),just as they are in the case when a, b, and c are real numbers. 29. Let b and c be complex numbers. Show that the roots of the quadratic equation 22 + bz + c 0 are complex conjugates of each other if and only if the quantity b2 - 4c is real and negative, b is real, and c is positive. 30. Let A be a complex number and B a real number. Show that the equation 4B. If this is so, show 221Re(Az)B-0has a solution if and only if|A that the solution set is a circle or a single point. 31. Let C be a circle and let A and B be any two distinct points on C. Show that if P is selected on the smaller arc of C joining A to B, then the angle from the segment AP to the segment BP is independent of P. This angle is π/2 if A and B are on opposite ends of a diameter. The result remains true if smaller is

32. Let z1,..., z, be complex numbers. Show by mathematical induction that Translation and Scaling* 33. Let C be a circle or a straight line. Show that the same is true of the locus of points z + β, z C, and β a fixed complex number. 34. Let C be a circle or a straight line. Show that the same is true of the locus of points az, z ε C, and α a fixed nonzero complex number. Inversion* 35. Let L be the line y-a, a >0. Show that the locus of points 1z, z e L, is the 36. Let L be a line through the origin. Show that the locus of points 1/z, z e L, is a 37. Let C be the circlelz- e0 e.Show that the locus of points 1/z,z e C, 38 Let C be the circle Iz-rl = r, r > 0. Show that the locus of points l/z, z e c, is circle of radius 1/2a centered at -i/2a. line through the origin. What is the relationship of the slopes of the two lines? is the circle centered at c/(c22), of radius r/c2 -r2). the vertical line through 1/2r.

Solution by an expert tutor
Blurred Solution
This question has been solved
Subscribe to see this solution