atom's Spesh Thread

[atom]

Hi everyone, I need help with question 29b) ii); I got the answers for 29a and 29bi but if someone could also show me the working for 29bi that would be great.

29) A complex number z satisfies the inequality |z + 2 − 2 sqrt3i| ≤ 2.
a) Sketch the corresponding region representing possible values of z.
b)
i) Find the least possible value of |z|.
ii) Find the greatest possible value of Arg z.

Thanks in advance!

[kinslayer]

For b) i), draw a line between the centre of the region in a) and the origin. It is perpendicular to the circle and it passes through the origin, so it is the shortest distance between them.

The part of the line between the centre and the edge of the circle has length 2, and the length of the whole line is 4 so the minimum value for is 2.

For part b) ii) the maximum/minimum arguments will be the arguments of the tangents to the circle that pass through the origin. The minimum is clearly . To get the maximum, use symmetry to reflect on the ray Arg z = 2pi/3 to get the maximum which is 2pi/3 + (2pi/3 - pi/2) = 5pi/6. I used MSPaint to draw a shitty diagram showing similar triangles which may help.