Answer to Question #118126 in Complex Analysis for Amoah Henry

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) y2 = 4x

Answer 1

$w=u+iv=z+4$

(i) $|z|=3$

$w=z+4=>z=w-4=>|z|=|w-4|=3$

The graph is the circle with center $(4, 0)$ and radius $3.$

$u=x+4, v=y$

$(u-4)^2+v^2=9$

(ii) $arg(z)=\dfrac{\pi}{3}$

$\dfrac{y}{x}=\tan(\dfrac{\pi}{3})=\sqrt{3}=>y=\sqrt{3}x,x>0$

$u=x+4, v=y$

$v=\sqrt{3}(u-4),u>4$

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 $5.$

(iv) $y^2=4x$

$u=x+4, v=y$

$v^2=4(u-4)$

The graph is the horizontal parabola.