x.Exhaust.x
Retired Member
1a) Let OABC be a square in the complex plane where O is the origin. The points A and C represent complex numbers z and iz respectively. Find the complex number which is presented by the point B.
b) This square is now rotated about O through 45 degrees in an anticlockwise direction to OPQR. Find the complex numbers that are represented by the points P, Q, and R.
2) In the attachment (see attachment), the complex numbers z1 and z2 are represented by the points P and Q respectively. Furthermore, the triangle OPQ is right isosceles with angle POQ = 90 degrees.
Show that (z1)^2+(z2)^2=0
3) Let w=(z-i)/(z+i). Find the locus of w given that the locus of z is:
a) The straight line Im(z)=1
b) The half-plane Im(z) greater than or equal to 0.
4) The diagram shows the locus of z such that arg(z+1)-arg(z+5)=pi/4 (see attachment)
This locus is part of a circle. The angle between the lines from (-5,0) to any point on the locus and from (-1,0) to the same point on the locus is equal to a constant, theta, as shown.
a) Justify the claim that theta = pi/4.
b) Find the centre of the circle.
5) Sketch: arg((z-4)/(z-5i))=pi/2
6) Let w=(az+b)/(cz+d)where a, b, c,and d are real cnstants with ad-bc > 0 and assume that the locus of z is given by the region Im(z) > 0. Show that w has the same locus as z.
7) The points representing the complex numbers z1, z2, and z3 lie on a circle that does not pass through the origin. Prove that the points representing 1/z1, 1/z2, and 1/z3 also lie on a circle.
And final note: What's the best way to remember subsets? Is there a list? Thanks.
b) This square is now rotated about O through 45 degrees in an anticlockwise direction to OPQR. Find the complex numbers that are represented by the points P, Q, and R.
2) In the attachment (see attachment), the complex numbers z1 and z2 are represented by the points P and Q respectively. Furthermore, the triangle OPQ is right isosceles with angle POQ = 90 degrees.
Show that (z1)^2+(z2)^2=0
3) Let w=(z-i)/(z+i). Find the locus of w given that the locus of z is:
a) The straight line Im(z)=1
b) The half-plane Im(z) greater than or equal to 0.
4) The diagram shows the locus of z such that arg(z+1)-arg(z+5)=pi/4 (see attachment)
This locus is part of a circle. The angle between the lines from (-5,0) to any point on the locus and from (-1,0) to the same point on the locus is equal to a constant, theta, as shown.
a) Justify the claim that theta = pi/4.
b) Find the centre of the circle.
5) Sketch: arg((z-4)/(z-5i))=pi/2
6) Let w=(az+b)/(cz+d)where a, b, c,and d are real cnstants with ad-bc > 0 and assume that the locus of z is given by the region Im(z) > 0. Show that w has the same locus as z.
7) The points representing the complex numbers z1, z2, and z3 lie on a circle that does not pass through the origin. Prove that the points representing 1/z1, 1/z2, and 1/z3 also lie on a circle.
And final note: What's the best way to remember subsets? Is there a list? Thanks.
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