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Question:

If |z + 2i| ≤ 4 then the difference between the greatest value and the least value of |z| is

2√113

2√13

4+√13

8

Solution:

given equation represents the circle with center(0, -2) and is of radius(R)=4
|z| represents the distance of point 'z' from origin
Greatest and least distances occur along the normal through the origin
Normal always passes through center of circle
From figure; let PQ be the normal through origin 'O' and C be its center(0, -2)
it is clear that OP is the least distance and OQ is the greatest distance
From diagram; OP = CP - OC and OQ = CQ + OC
Here, CP = CQ = R = 4
OC = √(0² + (-2)²) → OC = √4 = 2
∴OP = CP - OC → OP = 4 - 2 = 2
∴Least distance OP = 2
and OQ = CQ + OC → OQ = 4 + 2 = 6
∴Greatest distance = OQ = 6
Difference between greatest and least distance = OQ - OP = (6) - (2) → Difference = 4
However, if the question meant |z - 3 + 2i| ≤ 4, then:
given equation represents the circle with center(3, -2) and is of radius(R)=4
|z| represents the distance of point 'z' from origin
Greatest and least distances occur along the normal through the origin
Normal always passes through center of circle
From figure; let PQ be the normal through origin 'O' and C be its center(3, -2)
it is clear that OP is the least distance and OQ is the greatest distance
From diagram; OP = CP - OC and OQ = CQ + OC
Here, CP = CQ = R = 4
OC = √(3² + (-2)²) → OC = √13
∴OP = CP - OC → OP = 4 - √13
∴Least distance OP = 4 - √13
and OQ = CQ + OC → OQ = 4 + √13
∴Greatest distance = OQ = 4 + √13
Difference between greatest and least distance = OQ - OP = (4 + √13) - (4 - √13) → Difference = 2√13
final answer = 2√13
the correct option is 'B'