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If a complex number z lies on the interior or on the boundary of a circle of radius 3 units and centre (-4,0) find the greatest and least values of $\;|z+1| \;.$

$(a)\;3 \; and\;0\qquad(b)\;6\;and\;0\qquad(c)\;4\;and\;1\qquad(d)\;5\;and\;1$

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Answer : $\;6\;and\;0$
Explanation :
Distance of the point representing z from the centre of the circle is $\;|z-(-4+i0)| = |z+4|\;.$
According to given condition $\; |z+4| \leq 3$
Now $\;|z+1| = |z+4-3| \leq |z+4| + |(-3)| \leq 3+3 =6$
Therefore , the greatest value of $\;|z+1| \;$is 6
Since least value of the modulus value of a complex number is zero , the least value of $\;|z+1| =0\;.$
answered Apr 17, 2014 by yamini.v

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