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If z(1),z(2) and z(3), z(4) are two pair...

If `z_(1),z_(2)` and `z_(3), z_(4)` are two pairs of conjugate complex numbers, the find the value of `arg(z_(1)/z_(4)) + arg(z_(2)//z_(3))`.

Text Solution

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Let `z_(1) = r_(1)(costheta_(1) + I sintheta_(1))`,
Then, `z_(2) = barz_(1) = r_(1) (costheta_(1)- i sintheta_(1)) = r_(1)[cos(-theta_(1))+sin(-theta_(1))]`
Also, let `z _(3) = r_(2)(costheta_(2) + isintheta_(2))`,
Then, `z_(4) = barz_(3) = r_(2) (costheta _(2)-isin theta_(2))`
arg `((z_(1))/(z_(4))) + arg ((z_(2))/(z_(3))) = arg (z_(1))-arg(z_(4)) +arg(z_(2))-arg(z_(3))`
`=theta_(1)-(-theta_(2)+(-theta_(1))-theta_(2)" "[:.arg (z)=theta] `
`=theta_(1)+theta_(2)-theta_(1)-theta_(2) =0`
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