IIT JEE , AIEEE Forum - Mathematics , Algebra

neeraj_agarwal_1990

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

ans-4

shouldn't it be 0?

(using triangle inequality)

netkid07

this is not a standard method....but still works for some probs :)

it'll seem lame...but tryin it won't harm :)

|z-2|+|z-3|+|2z-9|

first put z=0....it becomes...2+3+9...so biggest contribution is of the last term....

so now....2z-9=0...z=9/2...

put this value...u get...2.5+1.5 = 4....

hope it helps...cheero :)

neeraj_agarwal_1990

this didn't help....couldn't understand the method...

netkid07

well, it's not a method actually.......just an observation........the expression tells about the addition of distances of a point z from... 2 , 3 , 4.5.......putting z at 4.5 wud compell the expression to take minimum value.....draw it on a number line...n den u'll be able to understand....

spideyunlimited

|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

feynmann

It's simple !!

| z-2 | + | z -3 | + | 2z - 9 |

= | z-2 | + | z -3 | + | 9 - 2z |

>= | z -2 + z - 3 + 9 - 2z | = 4

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