Question

Show that \( \frac{1-2 i}{3-4 i}+\frac{1+2 i}{3+4 i} \) is real

Answer 1

\(\frac {1- 2i}{3 - 4i} + \frac {1 + 2i}{3 + 4i}\)

\(= \frac {(1 - 2i) (3 + 4i) + (3 - 4i)(1 +2i)}{(3 + 4i)(3 - 4i)}\)

\(= \frac {3 + 4i - 6i - 8i^2 + 3 +6i -4i -8i^2}{9-16i^2}\)

\(= \frac {6 -16i^2}{9-16i^2}\)

\(= \frac {6 -16(-1)}{9-16(-1)}\)    ...[∵ i² = – 1]

\(= \frac {6 +16}{9 + 16}\) 

\(= \frac {22}{25}\), which is a real number.