Complex Numbers
Complex numbers satisfying modulus and linear constraints
MJAT_TS4_P1
Grade 12
Question:
The complex numbers $\omega$ and $z$ satisfy $|\omega|=5$, $|z|=13$, and $52\omega-20z=3(4+7i)$. The value of the product $\omega z = a+ib$ (where $a,b\in\mathbb{R}$). Then $a+b=$
Step-by-Step Solution
Key Concept: From $52\omega-20z=12+21i$: write $\omega$ and $z$ in terms of their real and imaginary parts. Together with $|\omega|=5$ and $|z|=13$ (both Pythagorean triples: $5=|3+4i|$ or $|4+3i|$, $13=|5+12i|$ or $|12+5i|$).
After solving the system: $\omega z = a+ib$ with $a+b=\mathbf{23}$.
Correct Answer: 23