[Math] Complex numbers – finding minimum value

Question

For all complex numbers $z_1,z_2$ satisfying $|z_1|=12$ and $|z_2-3-4i|=5$ , find the minimum value of $|z_1-z_2|$

Can we go like this :

Let $z_1 = x +iy$ therefore $|z_1| = \sqrt{x_1^2+y_1^2}$ and $z_2 = x_2+ iy_2$

$|z_2-3-4i| = \sqrt{(x_2-3)^2+ (y_2-4)^2}$ so $x_2^2-6x_2+9+y_2^2-8y_2+16 =25$ ( squaring both sides) ..
Please guide if it is correct..

Answer 1

Another hint:

Hence, the maximum is $22$ and the minimum is $2$ but you should prove this.