Question #118126
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
1
Expert's answer
2020-05-26T20:05:42-0400

w=u+iv=z+4w=u+iv=z+4

(i) z=3|z|=3

w=z+4=>z=w4=>z=w4=3w=z+4=>z=w-4=>|z|=|w-4|=3

The graph is the circle with center (4,0)(4, 0) and radius 3.3.

u=x+4,v=yu=x+4, v=y

(u4)2+v2=9(u-4)^2+v^2=9

(ii) arg(z)=π3arg(z)=\dfrac{\pi}{3}

yx=tan(π3)=3=>y=3x,x>0\dfrac{y}{x}=\tan(\dfrac{\pi}{3})=\sqrt{3}=>y=\sqrt{3}x,x>0

u=x+4,v=yu=x+4, v=y

v=3(u4),u>4v=\sqrt{3}(u-4),u>4

The graph is the ray excluding start point (4,0)(4,0)

(iii) z+4=5|z+4|=5

w=5|w|=5

The graph is the circle with center (0,0)(0,0) and radius 5.5.

(iv) y2=4xy^2=4x

u=x+4,v=yu=x+4, v=y

v2=4(u4)v^2=4(u-4)

The graph is the horizontal parabola.



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