Question

Complex Numbers Question Help Please

Answer 1

\[Given: z=\frac{ 1 + i }{ 3 + \alpha i }+\frac{ 3 + 2i }{ \alpha + 3i } and: \Re(z) = \Im(z) \]
find alpha where alpha is a real number

Answer 2

Suppose \[\large z = \frac{x_{1}+iy_{1}}{x_{2}+iy_{2}} \text{ multiply both numerator and denominator with}\] \(\large \text{the conjugate of }x_{2}+iy_{2} \text{ i.e. } x_{2}-iy_{2}\)
\[\large z = \frac{(x_{1}+iy_{1})(x_{2}-iy_{2})}{x_{2}^{2} + y_{2}^{2}} = \frac{x_{1}x_{2}+y_{1}y_{2} + i(y_{1}x_{2} - x_{1}y_{2})}{x_{2}^{2}+y_{2}^{2}}\]

Use the above information to find the real part and imaginary part of \(z\), and then equate them and form an equation which you can solve for \(\alpha\)