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Question

For complex numbers z1, z2 if z1=12 and z2-3-4i=5 then minimum value of z1-z2 is


A

0

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B

2

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C

7

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D

17

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Solution

The correct option is B

2


Step 1. Given Data,

z1=12

z2-3-4i=5

We have to calculate the minimum value of z1-z2.

Step 2. We have to find z1-z2,

Let z1=r1cosθ+isinθand z2=r2cosϕ+isinϕ.

Hence,

z1=r1cosθ2+r1sinθ2z1=r1z2=r2cosϕ2+r2sinϕ2z2=r2

z1-z2=r1cosθ+isinθ-r2cosϕ+isinϕz1-z2=r1cosθ-r2cosϕ+ir1sinθ-r2sinϕz1-z2=r1cosθ-r2cosϕ2+r1sinθ-r2sinϕ2z1-z2=r12(cos2θ+sin2θ)+r22(cos2ϕ+sin2ϕ)-2r1r2cosθcosϕ+sinθsinϕz1-z2=r12+r22-2r1r2cos(θ-ϕ)

As cos(θ-ϕ)=1if θ=ϕ

But we have, cos(θ-ϕ)1

z1-z2r12+r22-2r1r2z1-z2r1-r22z1-z2r1-r2z1-z2z1-z2

Step 3. Calculate the minimum value of z1-z2

We have, z1-z2z1-z2

Now, we can write z2-3-4ias,

z2-3-4iz2-3+4iz2-3-4iz2-32+42

Given that, z2-3-4i=5

Therefore,

z2-3-4iz2-32+425z2-32+425z2-5z210

Minimum value of z1-z2z1-z2

Given that, z1=12

z1-z212-10z1-z22

Hence, the correct option is 'B'.


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