Question
The greatest and the least absolute value of $$z + 1,$$ where $$\left| {z + 4} \right| \leqslant 3$$ are respectively
A.
6 and 0
B.
10 and 6
C.
4 and 3
D.
None of these
Answer :
6 and 0
Solution :
We have, $$\left| {z + 1} \right| = \left| {z + 4 - 3} \right|\,\,\,.....\left( {\text{i}} \right)$$
Now, $$\left| {z + 4 - 3} \right| \leqslant \left| {z + 4} \right| + \left| { - 3} \right| \leqslant 3 + 3 = 6$$
$$\left[ {{\text{Given }}\left| {z + 4} \right| \leqslant 3\,\,\& \,\,\left| { - 3} \right| = 3} \right]\,\,\therefore \left| {z + 1} \right| \leqslant 6$$
Again $$\left| {z + 1} \right| \geqslant 0$$ [modulus is always non-negative]
∴ Least value of $$\left| {z + 1} \right|$$ may be zero, which occurs
when $$z = - 1,$$ For $$z = - 1,\left| {z + 4} \right| = \left| { - 1 + 4} \right| = 3$$
Which satisfies the given condition that $${\left| {z + 4} \right| \geqslant 3}$$
Hence, the least and the greatest values of $$\left| {z + 1} \right|$$ are 0 and 6.