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Question

The locus of complex number z satisfying arg[z(1+i)]=⎪ ⎪⎪ ⎪3π4; |z||z2|π4; |z|>|z2| is

A
a straight line passing through (2,0).
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B
a straight line passing through (2,0),(1,1).
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C
is a line segment.
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D
is a set of two rays.
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Solution

The correct option is D is a set of two rays.
Given equation is arg[z(1+i)]=⎪ ⎪⎪ ⎪3π4; |z||z2|π4; |z|>|z2|
Let z=x+iy, then
When |z||z2|
x2+y2(x2)2+y204x+4x1
When |z|>|z2|
x>1

Now,
arg[z(1+i)]=⎪ ⎪⎪ ⎪3π4; Re(z)1π4; Re(z)>1


Therefore, the locus is a set of two rays.

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