Question

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) y = 4x

Answer 1

$z = x+iy, w=u+iv.$

(a) $w=z+4 \Rightarrow u+iv=(x+4)+iy,$ so $u=x+4$ and $y=v$ .

(i) If $|z|=3,$ then $\sqrt{x^2+y^2}=3$ and points $(x,y)$ are situated on a circle with center (0,0) and radius 3. Next, we should make a shift of this circle on a distance of 4 along x-axis, because $u=x+4$ . So for (u,v) we get a circle with center (4,0) and radius 3.

(ii) arg z if the angle between radius-vector of z and x-axis. Therefore, all (x,y) points are situated on the ray with initial point (0,0) and angle between the ray and x-axis if π/3. Strictly saying, point (0,0) doesn't belong to the ray because we can't define arg of a z=0.

We may describe this ray by the following conditions:

$x>0, \;\; y = x\tan\dfrac{\pi}{3} = \sqrt{3}x.$

To obtain the locus of (u,v) we should shift this ray along x-axis on a distance of 4, so the initial point (4,0) also does not belong to our ray. Therefore,

$u=x+4, \;\; v=y = \sqrt{3}x = \sqrt{3}(u-4)$ .

We may note, that argument of w will not be equal to π/3 in every point.

The picture of (u,v) is a kind of this sketch:

(iii) Let us determine the locus of z such as |z+4|=5.

$(x+4)^2+y^2= 5^2,$ so it's a circle with center in (-4,0) and radius 5. We should make a shift of this circle on a distance of 4 along x-axis, because $u=x+4,$ therefore (u,v) are located on circle with center in (0,0) and radius 5.

(iv) If we have an equation $y=4x$ , therefore points (x,y) are situated on a straight line inclined by an angle of $\arctan 4$ to the x-axis. This line passes through (0,0), so we may say that for points z

$\arg z = \arctan 4 \quad (\mathrm{for\, x \ge 0})$ or $\arg z = \pi + \arctan 4 \quad (\mathrm{for\, x \lt 0})$ .

Next, we know that $u=x+4$ and $v=y,$ so the locus of (u,v) can be defined as

$v=y=4x=4(u-4) \Rightarrow v = 4(u-4).$ It is also an equation of a straight line and the sketch is

We may also note, that argument of w will not be equal to $\arctan 4$ in every point.