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In the complex plane, the graph of \(\left | z-3\right |=2\left | z+3 \right |\) intersects the graph of  \(\left | z\right |=k\) in exactly one point. Find all possible values of \(k.\)

Enter all possible values, separated by commas.

 

\(z \) is a complex number by the way.

 Jul 20, 2020
 #1
avatar+1035 
+9

Here is a hint:

Change z to a+bi. Since the magnitude is the square root of a^2+b^2, solve for that. 

To make myself more clear, let me show you how to do the left side, and then you can to the right side.

 

Once we have changed z to a+bi, we can subtract 3 from a. This will give us (a-3)+bi.

Now, squaring a and b, we get

a^2-6a+9+b^2 for the inside of the square root. Therefore, the left side is \(\sqrt {a^2-6a+9+b^2}\).

 

Now, try to do the right side, and solve the equation.

 Jul 21, 2020
 #2
avatar+30929 
+2

Here's a bigger hint:

 Jul 21, 2020
 #3
avatar+146 
0

Thank you both; it really helped me.

Firebolt  Jul 22, 2020

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