I came across this interesting question,it is as follows:
a complex number $z$ satisfies the equation $|z^2-9| + |z^2| = 41$ and we are required to find the locus of $z$ and maximum value of |z|
my approach:
one of the easiest but very lengthy method is $z= x +iy$ putting this into the equation, squaring both sides and equating real and imaginary parts = 0, I finally obtained $|z+3| + |z-3| = 10$ and it is easy to see that $\max (|z|) = 5$.
however we wanted to know if there is an easier and more convenient method to solve this question .kindly help us out.
The triangle inequality for complex numbers states that $|z_1+z_2| ≤ |z_1| + |z_2|$. In your case $41 = |z^2-9| + |z^2| ≤ |z^2-9+z^2|$.