Complex numbers revision

Algebra Level 4

Find the complex number z z such that ∣ z − 2 + 2 i ∣ ≤ 1 \left| z-2+2i \right| \le 1 and z z has the least absolute value.

The answer is of the form: ( a − 1 b ) ( c − i ) \left( a-\frac { 1 }{ \sqrt { b } } \right) \left( c-i \right)

Find a + b + c a+b+c .

where b b is not a perfect square, and i = − 1 i = \sqrt { -1 }

Try THIS


The answer is 5.

Aditya Kumar by Aditya Kumar , 22, India

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2 solutions

Boon Yang
Apr 5, 2016

First sketch the complex number on the Argand diagram. The circle drawn can be represented with the equation, (x-2)^2 + (y+2)^2= 1, where (x, y) represents (Re(z), Im (z)). Since the complex no. with the least modulus lies on the line, y= -x, solve the simultaneous equations involving these two equations and compare a,b and c with the complex no. a, b=2, c=1 Hence, 2+2+1=5.

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Aditya Kumar
May 26, 2015

I'm sorry I couldn't post the complete solution as had to go out urgently.

The answer is: ( 2 − 1 2 ) ( 1 − i ) \left( 2-\frac { 1 }{ \sqrt { 2 } } \right) \left( 1-i \right)

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