Finding the locus represented by complex variable equations?

Question

I'm trying to solve these two problems related to complex number but hardly found a solution. I hope that someone can solve these and clear it up for me. Thank you.

1. |z+2|=2|z-1|
2. |z+5|-|z-5|=6

Answer 1

The first question was essentially answered in a comment: the locus is a circle.

So we deal with the second question. Consider the equation $$|z+5|-|z-5|=6.\tag{1}$$ Flip it over. We get $$\frac{1}{|z+5|-|z-5|}=\frac{1}{6}.$$ Multiply top and bottom by $|z+5|+|z-5|$. Setting $z=x+iy$, and simplifying a bit, we get $$|z+5|+|z-5|=\frac{10x}{3}.\tag{2}$$ Note that this forces $x$ to be positive. Now "add" (1) and (2), and simplify a bit. We get $$|z+5|=3+\frac{5x}{3}.\tag{3}$$ Finally, replace $|z+5|$ by $\sqrt{(x+5)^2+y^2}$, and square both sides. It all turns out very nice, we get the equation $\dfrac{x^2}{9}-\dfrac{y^2}{16}=1$. However, we only get the right-half branch of the hyperbola, because of the positivity constraint on $x$ noted earlier.

Remarks: $1.$ We can also write the given equation as $$\sqrt{(x+5)^2+y^2}= \sqrt{(x-5)^2+y^2}+6,$$ and square both sides. We get some nice cancellation, with a surviving "square root" term involving $12\sqrt{(x+5)^2+y^2}$. Bring this to one side, the rest tot the other side, and square again. We end up with the equation of a hyperbola.

$2.$ One could also do it without calculation, by recalling one definition of the hyperbola as the locus of points the difference of whose distances from $2$ distinct points is constant.