Smallest Argument of Complex Number z |z+8i|=4

[kishtik]

What is the argument of the complex number z which has the smallest argument in |z+8i|=4?

I solved the problem correctly but my answer is rather long
|z-(-8i)|=4 (drawing)
z_0=a+bi, z_1=-a+bi
|a+bi+8i|=4
sqrt(a^2+(b+8)^2)=4 (radius)
a^2+b^2+16b+64=16 (eq 1)
And from 4-8-sqrt(48) triangle
sqrt(a^2+b^2)=sqrt(48)
a^2+b^2=48 (eq 2)
Place into eq 1
48+16b+64=16
16b=-96
b=-6
Placing into eq 2
a^2+36=48
a^2=12
a=+-sqrt(12)
z_0=sqrt(12)-6i
z_1=-sqrt(12)-6i
From the drawing, the smallest argument is at the third zone.
tan theta=-6/-sqrt(12)
=sqrt(3)
so theta=240 degrees.

There must be a shorter solution, can you please help me?

 

[HallsofIvy]

Saying that |z+8i|= 4 is the same as saying (geometrically, in the complex plane) that the distance from z to -8i is 4. That is all z satisfying that are on a circle with center at -8i and radius 4.

The argument of a complex number is the angle the line from 0 to the number makes with the real axis. In this case it is geometrically clear that that will happen when the line from 0 to the circle is tangent to the circle (in the first quadrant). That tangent line, the line from 0 to -8i and the line from -8i to the point on the circle make a right triangle with hypotenuse of length 8 and one leg of length 4. (edited: I just realized that the problem only asks for the argument, not for the actual number itself.)

The angle from the (negative) imaginary axis is given by sin(φ)= 4/8= 1/2 and so φ= 30 degrees. The argument is 270- 30= 240 degrees just as you got.

 

[kishtik]

Originally posted by HallsofIvy
Saying that |z+8i|= 4 is the same as saying (geometrically, in the complex plane) that the distance from z to -8i is 4. That is all z satisfying that are on a circle with center at -8i and radius 4.

The argument of a complex number is the angle the line from 0 to the number makes with the real axis. In this case it is geometrically clear that that will happen when the line from 0 to the circle is tangent to the circle (in the first quadrant). That tangent line, the line from 0 to -8i and the line from -8i to the point on the circle make a right triangle with hypotenuse of length 8 and one leg of length 4. (edited: I just realized that the problem only asks for the argument, not for the actual number itself.)

Knew these.

The angle from the (negative) imaginary axis is given by sin(φ)= 4/8= 1/2 and so φ= 30 degrees. The argument is 270- 30= 240 degrees just as you got.

BLAAAAH! 4*2=8 I should have seen that!