Alg1131W8T3 - Graphing regions in the complex plane

Saturday, 21 April 2018

11:24 AM

Machine generated alternative text: When graphing regions in the complex plane, we use the notion of modulus and argument frequently. Recall that the modulus of z = a + bi is Izl = X/'NZÄY. This is the length of the vector from the origin 0 to z, or alternatively the distance from O to z. Since z — w is the complex number which corresponds to the vector from w to z, Iz — WI is the distance benveen the complex numbers w and z. The argument of z = a + bi is the angle 9 that the vector from 0 to z makes with the positive x-axis, and is denoted 9 = arg(z). Since going around once adds an angle of 2K, the argument of z is only defined up to 2K. Sometimes it is convenient to restrict the argument to be in the range in which case we use the notation Arg (z) and call this the principal argument of z. Match up the regions with the correct descriptions of them: i) {z e C : Iz - 31 < 2} is the region ii) {z e C : 1m z > 4} is the region Iz — 2il} is the region iv) arg 1m 3i 2i o -2i 2 3 4 6 7 Re 8

 

Machine generated alternative text: 1m 2i Regions in the complex plane; 80s style! Identify the regions in the complex plane (larger version available here)

 

Machine generated alternative text: 0 0 arg(z) C) 0 arg(z) + (1 + i) < @ 0 arg(z— 1 — i) < O Re(z) — Im(z) > 0 O Im(z) — Re(z) > 0 @ Im(z) + Re(z) < 0 O Re(z) — Im(z) 1.

 

Machine generated alternative text: 1m The blue shaded rectangle labelled A can be described as: @ 1.5 < Re(z) S 2 o < 1.5 0 1.5 < s 2 0 1.5 < < 2 The solid blue line labelled B can be described as: -2 and and -1 1 2 O Im(z) > 1 O Im(z) 2 1 C) 0 arg(z) C) 0 < arg(z) < @ 0 arg(z) 37T

 

Machine generated alternative text: x A classical construction of an ellipse is sometimes called the Gardiner's construction: a rope of fixed length k is fixed at two focal pants Fl and F2 The ellipse is then the locus (or path) of all the points X which make the rope taut So mathematically the condition is IXF11+lXF21=k Australian Rules Football (or AFL) is played on an elliptical field. The UNSW AFL ellipse; also known as the Village Green, may be viewed as part of the complex plane. It has focal points Fl = -20 = 20 + and 140 i) The length (goal to goal) of the Village Green is 140 ii) The width of the Village Green is (to the nearest metre) 134 iii) You have a ticket to watch UNSW vs USyd at the Village Green. Your seat number is 60 + 30i Are you inside the ellipse? yes. iv) Your friend has seat number 60 + 35i . Is your friend inside the ellipse? no Note: all measurements are in metres

 

Machine generated alternative text: 1m -2 -1 1 There is an analog of the Gardiner's construction for a hyperbola: the locus of a point X such that X Fl where Fl and F2 are fixed focal points and k is a fixed length. 2 —i Here are three hyperbolas. See if you can identify the hyperbolas in the picture (larger picture here) 1 is hyperbola A • liz-21 Iz — ill = 2 is hyperbola 1 is hyperbola 3 c

 

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