Let $$z_1$$ and $$z_2$$ be any two non-zero complex numbers such that $$3|z_1| = 2|z_2|$$. If $$z = \frac{3z_1}{2z_2} + \frac{2z_2}{3z_1}$$ then maximum value of $$|z|$$ is:

We are given two non-zero complex numbers $$z_1$$ and $$z_2$$ such that:

$$3|z_1| = 2|z_2| \implies \left| \frac{3z_1}{2z_2} \right| = 1$$

1. Define a new complex variable:

   Let $$u = \frac{3z_1}{2z_2}$$.

   From the given condition, we know that $$|u| = 1$$. This means $$u$$ can be represented in polar form as:

   $$u = e^{i\theta} = \cos\theta + i\sin\theta$$
2. Express $$z$$ in terms of $$u$$:

   The expression for $$z$$ is:

   $$z = \frac{3z_1}{2z_2} + \frac{2z_2}{3z_1}$$

   Substituting $$u$$, we get:

   $$z = u + \frac{1}{u}$$
3. Substitute the polar form:$$z = e^{i\theta} + e^{-i\theta}$$

   Using Euler's formula:

   $$z = (\cos\theta + i\sin\theta) + (\cos\theta - i\sin\theta)$$ $$z = 2\cos\theta$$
4. Find the Maximum Value of $$|z|$$:

   We need to find the maximum value of $$|z| = |2\cos\theta|$$.

   Since the range of the cosine function is $$[-1, 1]$$, the maximum value of $$|\cos\theta|$$ is $$1$$.

   $$|z|_{max} = 2(1) = 2$$

The maximum value of $$|z|$$ is 2.