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Question

Let z1=10+6i and z2=4+6i, where i=1. If z is any complex number such that arg(zz1zz2)=π4, then the value of |z79i| is

A
2
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B
3
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C
23
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D
32
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Solution

The correct option is D 32
Let z=x+iy
Then, zz1zz2=(x10)+i(y6)(x4)+i(y6)
=[(x10)+i(y6)][(x4)i(y6)](x4)2+(y6)2
=(x10)(x4)+(y6)2(x4)2+(y6)2+i6(y6)(x4)2+(y6)2

Now, arg(zz1zz2)=π4
6(y6)(x10)(x4)+(y6)2=1
x2+y214x18y+112=0 (1)

Now, |z79i|2=(x7)2+(y9)2
=x2+y214x18y+130
=18 [From (1)]
|z79i|=18=32

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