Further maths: Complex numbers and geometry

Risermax

8

Can someone explain to me question 8b? I replied to this with the question below.

(edited 11 months ago)

Reply 1

Risermax

OP

8

Original post by Risermax

Can someone explain to me question 8b?

Reply 2

mqb2766

21

The first set is obviously the ring from part a). That intersects with ...

If you consider when
|z-i| = |z-3i|
It means the distances between the points i and 3i are the same. So the perpendicular bisector of the line joining the points. Then the set
|z-i| <= |z-3i|
is given by the halfspace containing i (fairly trivially to sub in and check 0 <= 2. So the line y=2 (2i) seperates the two halfspaces and you want the lower one.

Then a bit of tedious trig.

(edited 11 months ago)

Reply 3

Risermax

OP

8

Original post by mqb2766

The first set is obviously the ring from part a). That intersects with ...

If you consider when
|z-i| = |z-3i|
It means the distances between the points i and 3i are the same. So the perpendicular bisector of the line joining the points. Then the set
|z-i| <= |z-3i|
is given by the halfspace containing i (fairly trivially to sub in and check 0 <= 2. So the line y=2 (2i) seperates the two halfspaces and you want the lower one.

Then a bit of tedious trig.

yh that part I understood, it’s the trig part im stick with. Why is the angles subtended at centre 2xcos^-1(1/2). I don't understand the segment area part either.

Reply 4

mqb2766

21

It should be ~gcse stuff that, but if you draw the diagram the line y=2 creates two
1:sqrt(3):2
right triangles so 30:60:90 so the angle of the relevant sector is 2*60 so youre finding the (shaded) area of a segment with angle 120 (1/3 of a circle) and radius 2. Then subtract that from the overall ring area.

The ms desription where they dont simply mention atan(sqrt(3)) = 60 is overly complex.