Question: Determine the curve in the complex plane that is determined by $ \lvert z-3\rvert + \lvert z+3\rvert =\lvert 10-3\rvert$
So far I have $ \lvert z-3\rvert + \lvert z+3\rvert =7$
$ \lvert (x+iy)-3\rvert + \lvert (x+iy)+3\rvert =7$ ; Since $z=x+iy$
However, now I do not know how to proceed with this problem. Any help will be greatly appreciated.
Geometrically speaking $$ \lvert z-3\rvert + \lvert z+3\rvert =\lvert 10-3\rvert$$ describes the set of point on the complex plane whose sum of distances from the points $3$ and $-3$ is a constant $7$
This is an ellipse with foci at $-3$ and $3$ with major diameter of $2a=7$
The equation of the ellipse is $$\frac {x^2}{a^2} + \frac {y^2}{b^2} = 1$$ where $a=3.5$ and $b= \sqrt {3.5^2-3^2}$