Let x, y, u, v be real numbers and z, w be complex numbers such that z = x + iy, w = u+iv. In each case, describe the locus of the point (u, v) in the Argand diagram.
(a) w = z + 4 and (i) |z| = 3, (ii) arg z = π/3, (iii) |z + 4| = 5, (iv) y2 = 4x
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Expert's answer
2020-05-26T20:05:42-0400
w=u+iv=z+4
(i) ∣z∣=3
w=z+4=>z=w−4=>∣z∣=∣w−4∣=3
The graph is the circle with center (4,0) and radius 3.
u=x+4,v=y
(u−4)2+v2=9
(ii) arg(z)=3π
xy=tan(3π)=3=>y=3x,x>0
u=x+4,v=y
v=3(u−4),u>4
The graph is the ray excluding start point (4,0)
(iii) ∣z+4∣=5
∣w∣=5
The graph is the circle with center (0,0) and radius 5.
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