9709 P31 - Nov 2023 - Q4
1961
The complex number u is defined by \(u = \frac{3 + 2i}{a - 5i}\), where a is real.
(a) Express u in the Cartesian form \(x + iy\), where x and y are in terms of a.
(b) Given that \(\arg u = \frac{1}{4}\pi\), find the value of a.
Solution
(a) To express \(u\) in Cartesian form, multiply the numerator and denominator by the conjugate of the denominator:
\(u = \frac{(3 + 2i)(a + 5i)}{(a - 5i)(a + 5i)}\)
Using \(i^2 = -1\), the denominator becomes:
\(a^2 + 25\)
Expanding the numerator:
\((3 + 2i)(a + 5i) = 3a + 15i + 2ai + 10i^2\)
\(= 3a + 15i + 2ai - 10\)
\(= (3a - 10) + (2a + 15)i\)
Thus, \(u = \frac{3a - 10}{a^2 + 25} + \frac{2a + 15}{a^2 + 25}i\).
(b) Given \(\arg u = \frac{1}{4}\pi\), we have:
\(\arctan\left(\frac{2a + 15}{3a - 10}\right) = \frac{\pi}{4}\)
This implies:
\(\frac{2a + 15}{3a - 10} = 1\)
Solving for \(a\):
\(2a + 15 = 3a - 10\)
\(25 = a\)
Thus, \(a = 25\).
