Q) Let z be a complex number ,then the minimum value of |z-2|+|z-3|+|2z-9| is ?

A) |z-2|+|z-3|+|2z-9|

Let z = a + ib


|z-2|+|z-3|+|2z-9|
= sqrt [ (a-2)^2 + b^2 ] + sqrt [ (a-3)^2 + b^2 ] + sqrt [ (2a - 9)^2 + 4b^2 ]

for minimising, b^2 will be 0.

= sqrt [ (a-2)^2 ] + sqrt [ (a-3)^2 ] + sqrt [ (2a - 9)^2 ]
= |a-2| + |a-3| + |2a - 9|
if seeing first two terms, a - 3> 0 then a > 3
then |2a - 9| will be 9 - 2a

= a - 2 + a - 3 + 9 - 2a

= 4

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