Question # If ( z ) is a complex number such that ( |z| geq 2 ), then the minimum value of ( left|z+frac{1}{2}right| )

# 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 )

Solution

( |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)