Complex Numbers
Complex Numbers
nta_abhyas_2025
Grade None
Question:
If $A(2 + 3i), B(3i)$ and $D(4i)$ are two vertices of a square $ABCD$ (taken in anticlockwise order) in a complex plane, then the value of $|Z_1|^2 - |Z_2|^2$ (where $C$ is $Z_1$ and $D$ is $Z_1$) is equal to
Step-by-Step Solution
Key Concept: Complex equations involving moduli and arguments can be solved by converting exponential form to rectangular form.
We have $\frac{z_1 - (2+3i)}{(3+4i)(2-3i)} = e^{i\pi/4} = 1$. Computing: $z_1 = 2 + 3i + (3+4i)(2-3i) = 2 + 3i + (6 - 9i + 8i - 12i^2) = 2 + 3i + 6 - i + 12 = 20 + 2i$. Similarly, for $z_2$: $\frac{z_2 - (2+3i)}{(3+4i)(2-3i)} = e^{-i\pi/4} = 1$ gives $z_2 = 2 + 3i + (3 + 4i)(2-3i)e^{-i\pi/4}$. After computation, $|z_1|^2 - |z_2|^2 = 20 - 17 = 12$.
Correct Answer: 12