Question

The complex number w is given by w= p-4i/2-3i where p is a real constant. Express w in the form a+bi where a and b are real constants. give your answer in simplest terms of p.b) given that arg w= pi/4 find the value of p

Answer 1

Step 1

Given;

$w=\frac{p-4i}{2-3i}$

Required; To express w in the form of a+bi.

Step 2

Express w in the form of a+bi

Multiply the numerator and the denominator by the binomial conjugate of (2-3i)

The binomial conjugate of (2-3i) = (2+3i)

$w=(\frac{p-4i}{2-3i})\times(\frac{2+3i}{2+3i})=\frac{2p+3pi-8i-12i^2}{4+6i-6i-9i^2}$$\begin{gathered} w=\frac{2p+3pi-8i-12(-1)}{4-9(-1)} \\ \text{Note; i}^2=(\sqrt[]{-1})^2=-1 \end{gathered}$$\begin{gathered} w=\frac{2p+3pi-8i+12}{4+9}=\frac{2p+3pi-8i+12}{13} \\ w=\frac{(2p+12)+(3p-8)i}{13} \\ w=\frac{(2p+12)}{13}+\frac{(3p-8)i}{13} \\ w=\frac{2(p+6)}{13}+\frac{(3p-8)i}{13} \end{gathered}$

where;

$\begin{gathered} a=\frac{2(p+6)}{13}_{} \\ bi=\frac{(3p-8)i}{13} \end{gathered}$