CameraIcon

SearchIcon

MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Properties of Argument

If z 1 and z ...

Question

$If z_{1} and z_{2}$ are two complex numbers such that $| z_{1} | = | z_{2} | and (z) + a r g (z_{2}) = π, then show that z_{1} = -_{2} .$

Open in App

Solution

$| z_{1} | = | z_{2} | Let a r g (z_{1}) = θ ∴ a r g (z_{2}) = π - θ In polar form, z_{1} = | z_{1} | (c o s θ + i s i n θ) . . . . (i) z_{2} = | z_{1} | (c o s (π - θ) + i s i n (π - θ)) = | z_{2} | (- c o s θ + i s i n θ) = - | z_{2} | (c o s θ - i s i n θ) Finding conjuage of_{2} = - | z_{2} | (c o s θ + i s i n θ) . . . (i i) (i) ∖ (i i) is equal to \frac{z_{1}}{_{2}} = - \frac{| z_{1} | (c o s θ + i sin θ)}{| z_{2} | (c o s θ + i sin θ)} \frac{z_{1}}{_{2}} = - \frac{| z_{1} |}{| z_{2} |} [∵ | z_{1} | = | z_{2} |] \frac{z_{1}}{_{2}} = - 1 z_{1} = -_{2}$

Hence proved.

Suggest Corrections

thumbs-up

1

Similar questions

Q. If z₁ and z₂ are two complex numbers such that

|z_{1}| = |z_{2}|

and arg(z₁) + arg(z₂) =

π

, then show that

z_{1} = - \bar{z_{2}}

z_{1}

z_{2}

are two complex numbers such that

z_{1} \neq z_{2}

| z_{1} | = | z_{2} |

\frac{z_{1} + z_{2}}{z_{1} - z_{2}}

z_{1}

z_{2}

are two complex numbers such that

| z_{1} | = | z_{2} |

a r g z_{1} + a r g z_{2} = π

z_{1}

z_{2}

z_{1}

z_{2}

are two non-zero complex numbers such that

| z_{1} - z_{2} | = | | z_{1} | - | z_{2} | |

arg (z_{1}) - arg (z_{2}) =

z_{1}

z_{2}

are two complex numbers such that

| z_{1} | = | z_{2} | + | z_{1} - z_{2} |

arg (z_{1}) - arg (z_{2})

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

De Moivre's Theorem

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Properties of Argument

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon