If z = x + iy is a complex number, then sketch the set of points that satisfy the following equations?

Question

i) /z − 2 + 3i/ ≥ 2
ii) −1 < Re(z) ≤ Im(z + i)

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Answer 1 · 2018-02-13T15:07:26

See below.

If #z = x + i y# then #bar z = x - i y# and

#sqrt(z cdot bar z) = abs(z)# so

given #w = z-2 + 3i# then #absw = sqrt(w cdot bar w) = sqrt(z-2+3i)(bar z-2-3i)) = sqrt(x^2+y^2-4x+6y+13)ge 2#

or squaring both sides

#x^2+y^2-4x+6y+13 ge 4# or

#(x-2)^2+(y+3)^2 ge 2^2#

In light blue the feasible region.

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Now regarding #-1 < "Re"(z) le "Im"(z+i)# we have

#-1 < x and x le y+1#

The corresponding feasible region follows

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