JEE PYQ: Complex Numbers Question 26
Question 26 - 2020 (09 Jan Shift 1)
Let $z$ be a complex number such that $\left|\frac{z-i}{z+2i}\right| = 1$ and $|z| = \frac{5}{2}$. Then the value of $|z + 3i|$ is:
(1) $\sqrt{10}$
(2) $\frac{7}{2}$
(3) $\frac{15}{4}$
(4) $2\sqrt{3}$
Show Answer
Answer: (2)
Solution
$|z-i| = |z+2i|$: locus is $\text{Im}(z) = -\frac{1}{2}$, i.e., $y = -\frac{1}{2}$. $|z| = \frac{5}{2}$: $x^2 + \frac{1}{4} = \frac{25}{4} \Rightarrow x^2 = 6$. $|z+3i| = \sqrt{6 + (3-\frac{1}{2})^2} = \sqrt{6 + \frac{25}{4}} = \sqrt{\frac{49}{4}} = \frac{7}{2}$.
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