Problem

4. $z_1 = p + 5i$, $z_2 = 9 + 8i$ and $z_3 = \dfrac{z_1}{z_2}$

where $p$ is a real constant.

(a) Determine $z_3$ in the form $x + iy$, where $x$ and $y$ are in terms of $p$

(b) Determine the exact value of the modulus of $z_2$

Given that the argument of $z_1$ is $\dfrac{\pi}{3}$

(c) (i) determine the exact value of $p$

(ii) determine the exact value of the modulus of $z_3$