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# If anyone could help I'd really appreciate it

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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
+1041
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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
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Here's a bigger hint:

Jul 21, 2020
#3
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Thank you both; it really helped me.

Firebolt  Jul 22, 2020