Question

Need help with Part A

Answer 1

$z=\frac{8+pi}{p-4i}$
given Re(z)=2/5

Part (a) to find the possible values of p.

1. we rationalize z by multiplying the denominator by it's conjugate
$z=\frac{8+pi}{p-4i}$
$=\frac{(8+pi)(p+4i)}{(p-4i)(p+4i)}$  ]multiple top & bottom by conjugate of denominator]
$=\frac{(8+pi)(p+4i)}{(p-4i)(p+4i)}$
$=\frac{4p+(p^2+32)i}{p^2+16}$
$=\frac{4p}{p^2+16}+\frac{(p^2+32)}{p^2+16}i$

2. The real part is therefore $=\frac{4p}{p^2+16}$
and we have been given that Re(z)=2/5.
We now form the equation
$\frac{4p}{p^2+16}=2/5$
which transforms to the quadratic equation
$2(p^2+16)-20p=0$
and simplifies to
$p^2-10p+16=0$
and factors to
$(p-8)(p-2)=0$
and using the zero product property, we deduce that

p=8 or p=2 

Check:
substitute p=8 in z gives $\frac{2}{5}+\frac{6i}{5}$  ....good
substitute p=2 in z gives $\frac{2}{5}+\frac{9i}{5}$  ....good

If you need help with the other parts, please let me know.