Complex Numbers
Locus and Maximum Modulus
nta_pyq_2024_apr
Grade 11
Question:
Let $z$ be a complex number such that the real part of $\dfrac{z-2i}{z+2i}$ is zero. Then, the maximum value of $|z-(6+8i)|$ is equal to:
Step-by-Step Solution
Key Concept: Real part of $\frac{z-2i}{z+2i}=0$ means $\frac{z-2i}{z+2i}+\overline{\left(\frac{z-2i}{z+2i}\right)}=0$. This simplifies to $|z|^2=4$, i.e. $z$ lies on circle of radius 2.
$|z|=2$. $|6+8i|=10$. Max $|z-(6+8i)|=10+2=12$.
Correct Answer: 1