Question

Complex numbers help. *question attached below* will give medal and fan

Answer 1

So I have already found the first part of this question, which is 1+2i but I am stuck at the second part. I know the diagram needs to show a circle centered at (1,2i), with a radius of sqrt5, but how do I find the greatest value of |z| subject to this condition?

Answer 2

Notice that the circle passes through (0,0)

Answer 3

and its center is (1,2)

Answer 4

since the radius is sqrt(5), the point (2,4) must also on circle, yes ?

Answer 5

How do we know that the point (2,4) be on the circle?

Answer 6

@ganeshie8

Answer 7

helloooo? anyone home?

Answer 8

its the diametrically opposite point to 0,0

0 -> 1 -> 2
0 -> 2 -> 4

or with some other logic :

\( \sqrt 5 = \sqrt {1^2 +2^2 } = \sqrt {2^2+1^2 }\)

so every point, with distance of 1 unit in x direction from x=1 (that is 0 and 2)
AND distance of 2 units in y direction from y=2 (that is 0 and 2) will lie on the circle
so we get 0,0 and 2,4 to lie on the circle.

Answer 9

Okay, I get that @hartnn

Answer 10

since 0,0 produce least value of |z|
2,4 will give you the greatest value! :)

Answer 11

Is there an equation or expression as to how we got 2,4?

Answer 12

|x+iy-1-2i| =sqrt 5
(x-1)^2 + (y-2)^2 = 5
thats your circle....

Answer 13

yes

Answer 14

not able to think of a mathematical way to get 2,4 or the greatest value of |z| ..yet

Answer 15

alrighty

Answer 16

Maximize x^2 + y^2 subject to (x-1)^2 + (x-2)^2 = 5

Answer 17

so do we make x^2+y^2 equal to (x-1)^2 + (x-2)^2 =5?

Answer 18

that works, but that would be like digressing from complex numbers
so lets think of something else thats related to complex stuff

Answer 19

Not setting them equal to each other,
you must have studied optimization in calculus ?

http://www.wolframalpha.com/input/?i=Maximize++x%5E2+%2B+y%5E2++subject+to++%28x-1%29%5E2+%2B+%28y-2%29%5E2+%3D+5

Answer 20

but i don't like this method, there should some other better way to work the point

Answer 21

I've actually never done optimization in calculus :O