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Question

Locus of z if |zz1|=|zz2|, where z1 and z2 are complex numbers with the greatest and the least moduli, is

A
Line parallel to the real axis
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B
Line parallel to the imaginary axis
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C
Line having a positive slope
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D
Line having a negative slope
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Solution

The correct option is B Line parallel to the imaginary axis
We have
|z|4zz4z=2

2|z|4|z|2

|z|2+2|z|40 and |z|22|z|40

(|z|+1)250 and (|z|1)25

(|z|+1+5)(|z|+15)0

and (|z|1+5)×(|z|15)0

|z|51 or |z|51 and 51|z|5+1

51|z|5+1

Hence, the least modulus is 51 and the greatest modulus is 5+1, Also,

|z|=5+14|z|=51

Now, 4z=4¯¯¯z|z|2

Hence, 4/z lies in the direction of ¯¯¯z.

z4z=PR=2 (given)

We have
OP=5+1 and OR=51

cos2θ=OP2+OR2PR22OPOR

=(5+1)2(51)242(51)=1

2θ=0,2π

θ=0,π

z is purely real

z=±(5+1)

Similarly for |z|=51, we have z=±(51).

350694_117303_ans.png

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