Cartesian equations for complex numbers

Question

I'm new to this site but I have seen some proof questions done which greatly helped me a lot during my studies. So I need some help with finding the Cartesian equation for the locus of points $P(x,y)$ if:

$$z=x+yi$$ and $$\vert z+3\vert+\vert z-3\vert=8$$

I need help in getting the answer to be: $$7x^2+16y^2=112$$ and I've tried substituting the $z=x+yi$ into the modulus part but I don't know where to go from there. So any help with this question is greatly appreciated! Thanks!

Answer 1

Remind the definition of ellipse: The set of points whose sum of distances to two points are the same. In this case, the sum of distances from $z$ to $(3, 0), (-3, 0)$ are 8, so the trajectory of $z$ is ellipse with the length of semimajor axis is 4. Since the distance between two foci is 6, the length of semiminor axis must be $1/2 \times \sqrt {8^2-6^2}=\sqrt 7$. Therefore, the Cartesian equation must be ${x^2}/16+{y^2}/7=1$, or equivalently, $7x^2+16y^2=112$.