Find the equation of the locus of the point representing $z$ if $|z – 3i| – |z + 3i| = 2$.
Hi guys , so I can find the locus algebraically plugging in $x+iy=z$ and then manipulating it but that is a very tedious method also my answer came out to $8y^2-x^2=8$ however I am wondering how I would interpret this geometrically without doing too much algebra..
The locus of points where the absolute value of the difference of the distances to two given points is a constant is a hyperbola. These two points are called foci. See here for the cartesian equation of the locus given the foci.
In your case $|z–3i|–|z+3i|=2$ means the difference of the distances of the point $z$ to the points $3i$ and $-3i$. No absolute value here. So it is not a whole hyperbola but only one branch. For example $-i$ (or $(0,-1)$) satisfies the equation $$|z – 3i| – |z + 3i| = 2$$ but $i$ (or $(0,1)$) does not. So the equation of the locus is not $8y^2-x^2=8$. It should be $$y=-\sqrt{1+\frac{x^2}{8}}.$$