If z is a complex number such that ...
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If ( z ) is a complex number such that ( |z| geq 2 ), then the minimum value of ( left|z+frac{1}{2}right| )
(a) is equal to ( 5 / 2 )
(b) lies in the interval ( 1,2 )
(c) is strictly greater than ( 5 / 2 )
(d) is strictly greater than ( 3 / 2 ) but less than ( 5 / 2 )

JEE/Engineering Exams
Physics
Solution
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If z is a complex number such that |z|≥2, then the minimum value of ∣z+1/2∣
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( |z| geq 2 ) is the region on or outside circle whose centre is
(0, 0) and radius is ( 2 . ) Minimum ( left|z+frac{1}{2}right| ) is distance of ( z, ) which lie on circle ( |z|=2 ) from ( (-1 / 2,0) )
( therefore ) Minimum ( left|z+frac{1}{2}right|= ) Distance of ( left(-frac{1}{2}, 0right) ) from (-2,0) ( =sqrt{left(-2+frac{1}{2}right)^{2}+0}=frac{3}{2}=sqrt{left(frac{-1}{2}+2right)^{2}+0}=frac{3}{2} )
Geometrically Min ( left|z+frac{1}{2}right|=A D )
Hence, minimum value of ( left|z+frac{1}{2}right| ) lies in the interval (1,2)

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