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Question

If z lies on the curve |z34i|=3, then least value of |z| is

A
2
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B
3
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C
8
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D
None of these
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Solution

The correct option is A 2
Let z=x+iy
Hence, |(x3)+(y4)i|=3
Therefore, (x3)2+(y4)2=9
Hence z lies on the above circle.
Let,
x3=3cosθ
x=3+3costheta
y4=3sinθ
y=4+3sinθ
Hence ,z=3(1+cosθ)+i(4+3sinθ)
|z|2=9(1+cosθ)2+(4+3sinθ)2 ...(i)
=9[1+cos2θ+2cosθ]+[16+24sinθ+9sin2θ]
=25+9+18cosθ+24sinθ
=34+18cosθ+24sinθ
Differentiating with respect to θ and equating with 0, we get
18sinθ+24cosθ=0
3sinθ=4cosθ
tanθ=43
Hence,
sinθ=±45

cosθ=±35
Now Considering
sinθ=45

cosθ=35

z=3(135)+i(4125)

65+i85
Hence
|z|min=105
=2

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