Subsets of the complex plane



Subsets of the complex planeModelled solutionsDetermine the Cartesian equation, describe and sketch the graph defined by:z=3Finding the Cartesian equation:Geometric interpretation:z=3 is the set of points 3 units from the origin. A circle with centre (0, 0) and radius of 3.x2+y2=9Algebraic interpretation:Let z=x+iy|(x+iy|=3x2+y2=3 (definition of the modulus)x2+y2=9The equation represents a circle with centre 0, 0 and radius 3 or the set of all points which are 3 units from the origin.Sketch the graph:|z-3i|≤4Finding the Cartesian equation:Geometric interpretation:z-3i≤4 This is of the form z-z1, where z1=0+3iThis is the set of points less than or equal to 4 units from the point (0, 3) or the set of all points on or inside the circle with centre (0, 3) and radius of 4.x2+(y-3)2≤16Algebraic interpretation:Let z=x+iy|(x+iy)-3i|≤4|(x+i(y-3)|≤4x2+(y-3)2≤4 (definition of the modulus)x2+(y-3)2≤16This is the set of points on or inside the circle with centre (0, 3) and radius of 4 or the set of all points less than or equal to 4 units from the point (0, 3)Sketch the graph:|z+3-2i|=5Finding the Cartesian equation:Geometric interpretation:z+3-2i=5 This is of the form z-z1, where z1=-3+2iThis is the set of points 5 units from the point (-3, 2) or a circle with centre (-3, 2) and radius of 5.(x+3)2+(y-2)2=25Algebraic interpretation:Let z=x+iyx+iy+3-2i=5x+3+iy-2=5(x+32+(y-2)2=5 (definition of the modulus)(x+3)2+(y-2)2≤25This is the set of points on the circle with centre (-3, 2) and radius of 5 or the set of all points 5 units from the point (-3, 2)Sketch the graph:π4≤Arg(z)≤3π4Geometric interpretation: π4≤Arg(z)≤3π4 is the set of all points which have an argument between π4 and 3π4 inclusiveAlgebraic interpretation:Let z=x+iyArgz=θ, where θ=tan-1yxπ4≤Argx+iy≤3π4tan-1yx≥π4yx≥1y≥xtan-1yx≤3π4yx≤-1y≥-x(Change the sign as x is negative as 3π4 is in the 2nd quadrant)This will represent the region which satisfies two conditions or the overlap in two regions. Region 1: y≥x: Area above and including the line y=x.Region 2: y≥-x: Area above and including the line y=-x.Sketch the graphRe(z)>Im(z)Algebraic interpretation:Let z=x+iyRez=x and Imz=yx>yy<xThis is the region where y<x, or the area below the line y=xSketch the graph|z-1|=2|z-i|Finding the Cartesian equation:Geometric interpretation:Set of points which are twice as far from (1, 0) as they are from (0, 1)Algebraic interpretation:Let z=x+iy|(x+iy)-1|=2|(x+iy)-i||(x-1)+iy|=2|x+i(y-1)|(x-1)2+y2≤2x2+(y-1)2 (definition of the modulus)(x-1)2+y2≤4x2+y-12 (squaring both sides)x2-2x+1+y2=4x2+y2-2y+1x2-2x+1+y2=4x2+4y2-8y+4-3=3x2+2x+3y2-8y3x2+2x+3y2-8y=-33x2+23x+3y2-83y=-3x2+23x+3y2-83y=-1x2+23x+19+y2-83y+169=-1+19+169x+132+y-432=89The equation represents a circle with centre -13, 43 and radius 3 or the set of points which are 3 units from the origin. Sketch the graphGeogerba applet demonstrating the solution where C is a moveable point and represents the solution.4Rez+3Imz=12Finding the Cartesian equation:Algebraic interpretation:Let z=x+iyRez=x and Imz=y4x+3y=12Ory=-43x+4This is a linear relationship withy-intercept of 4 and x-intercept of 3.Gradient -4/3 and y-intercept of 4.Sketch the graphArgz-3+2i=π3Geometric interpretation:Argz-3+2i=π3This is of the form Arg(z-z1) where z1=3-2iArgz-3+2i=π3 is the set of all points which lie of the vector from (3, -2) at an angle of π3 from the horizontal.Sketch the graph|z-2+i|+z-2=3Finding the Cartesian equation:Geometric interpretation:Set of points whose distance from (2, -1) and distance from (2,0) sums to 3.Algebraic interpretation:Let z=x+iyx+iy-2+i+x+iy-2=3x-2+iy+1+|(x-2)+iy)|=3(x-2)2+(y+1)2+(x-2)2+y2=3 (definition of the modulus)x2-4x+4+y2+2y+1+x2-4x+4+y2=3x2-4x+y2+2y+5+x2-4x+y2+4=3x2-4x+y2+2y+5+x2-4x+y2+4+2x2-4x+y2+2y+5x2-4x+y2+4=9 (squaring both sides)2x2-8x+2y2+2y+9+2x2-4x+y2+2y+5x2-4x+y2+4=9 2x2-4x+y2+2y+5x2-4x+y2+4=-2x2+8x-2y2-2y 4x2-4x+y2+2y+5x2-4x+y2+4=-2x2+8x-2y2-2y2 (squaring both sides)LHS=4x4-32x3+8x2y2+8x2y+100x2-32xy2-32xy-144x+4y4+8y3+36y2+32y+80RHS=4x4-32x3+8x2y2+8x2y+64x2-32xy2-32xy+4y4+8y3+4y2 100x2-144x+36y2+32y+80=64x2+4y236x2-144x+32y2+32y+80=09x2-36x+8y2+8y+20=0 9x2-4x+8y2+y=-209x2-4x+4+8y2+y+14=-20+36+29x-22+8y+122=18(divide both sides by 8×9=72)x-222+y+1229=14x-222+y+12294=1Ellipse with centre 2, -12 with minor axis of 2 and major axis of 32Sketch the graphGeogerba applet demonstrating the solution where C is a moveable point and represents the solution. ................
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