Let z and w be two non- zero complex numbers such that -z- - -w- and z - w- - Then the value of z - w10 is

Let (w)= θ. Then (z)= π θ. nbsp; ∴ w = |w| (cosθ + i sinθ) and z = |z| (cos(π θ) + isin(π θ)) = |w| ( cosθ + i sinθ) [∵ |z| = |w|] = |w| (cosθ i sinθ ...

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Byju's Answer

Standard XII

Mathematics

Properties of Argument

Let z and w b...

Question

Let $z$ and $w$ be two non- zero complex numbers such that $| z | = | w |$ and $arg (z) + arg (w) = π$ . Then the value of $(z + ¯ ¯¯ ¯ w)^{10}$ is

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Solution

Let $arg (w) = θ$ . Then $arg (z) = π - θ$ .
$∴ w = | w | (cos θ + i sin θ)$
and
$z = | z | (cos (π - θ) + i sin (π - θ))$
$= | w | (- cos θ + i sin θ) [∵ | z | = | w |]$
$= - | w | (cos θ - i sin θ) = - ¯ ¯¯ ¯ w ∴ z + ¯ ¯¯ ¯ w = 0$ .
$\Rightarrow (z + ¯ ¯¯ ¯ w)^{10} = 0$

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Let z and w be two non- zero complex numbers such that |z|=|w| and arg(z)+arg(w)=π. Then the value of (z+¯¯¯¯w)10 is

Let arg(w)=θ. Then arg(z)=π−θ. ∴w=|w|(cosθ+isinθ) and z=|z|(cos(π−θ)+isin(π−θ)) =|w|(- cosθ+i sinθ)[∵|z|=|w|] =−|w|(cosθ−i sinθ)=−¯¯¯¯w∴z+¯¯¯¯w=0. ⇒(z+¯¯¯¯w)10=0

Let $z$ and $w$ be two non- zero complex numbers such that $| z | = | w |$ and $arg (z) + arg (w) = π$ . Then the value of $(z + ¯ ¯¯ ¯ w)^{10}$ is