If z is a complex number such that |z| ≥ 1, then the minimum value of

Question

If \(\mathrm{z}\) is a complex number such that \(|\mathrm{z}| \geq 1\), then the minimum value of \(\left|\mathrm{z}+\frac{1}{2}(3+4 i)\right|\) is:

(1) \(\frac{5}{2}\)

(2) 2

(3) 3

(4) \(\frac{3}{2}\)

(5) 0

Answer 1

Correct option is (5) 0

\(|Z| ≥ 1\) is a region outside & periphery of circle with centre (0, 0) & radius = one unit

minimum value of \(\left| Z - \left( -\frac32 - 2i\right)\right|\) is minimum distance of p point in \(|Z| ≥ 1\) from \(\left( - \frac 32 -2\right)\)

Therefore minimum value of \(\left| Z + \frac 12(3 + 4i)\right| = 0\)