If z is any complex number satisfying | z - 3 - 2 i | 2 , then the minimum value of | 2 z - 6 + 5 i | is [2011]

Question

If $z$ is any complex number satisfying $| z - 3 - 2 i |$ $\leq 2$ , then the minimum value of $| 2 z - 6 + 5 i |$ is [2011]

Smart Achievers · Accepted Answer

(5)

$Given:$ $| z - 3 - 2 i |$ $< 2,$

which represents a circular region with centre $(3, 2)$ and radius 2.

$Now,$ $| 2 z - 6 + 5 i |$ $= 2$ $| z - (3 - \frac{5}{2} i) |$

$= 2 \times distance of z from P (where z lies in or on the circle)$

$Also min distance of z from P = \frac{5}{2}$

$∴$ $Minimum value of$ $| 2 z - 6 + 5 i |$ $= 5$

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