Math Is Fun Forum

Identity

Member

Registered: 2007-04-18

Posts: 934

Find maximum w

Please help me with this question,

Let

Find

such that

is a maximum in S.

Thanks

Last edited by Identity (2009-04-03 05:39:52)

whatismath

Member

Registered: 2007-04-10

Posts: 19

Re: Find maximum w

You might want to first consider the boundary of S:

which is the set of complex numbers whose distance from 0
is half of that from

.

In general, if

, and

are two complex numbers, then the solution to the equation

is a circle:
applying

and
a "completing square"-like operations, we can derive from
the above equation that

In the question,

and the circle has center

and radius

.

The inequality

gives the inside of the circle. The point farthest from 0 is one of the 2 endpoints
on the diameter along the line between 0 and the center

.
The distance between 0 and this center is 2 and the radius is 4. So maximum of

is 2+4=6.

Or we could see geometrically this

satisfies

, on the
line through 0 and

and on the other side of

with respect to 0. So

(simple arithmetic).

Last edited by whatismath (2009-05-11 17:57:20)