$arg\,(5 - \sqrt 3 i) = $
$arg\,(5 - \sqrt 3 i) = $
- A
${\tan ^{ - 1}}\frac{5}{{\sqrt 3 }}$
- B
${\tan ^{ - 1}}\left( { - \,\frac{5}{{\sqrt 3 }}} \right)$
- C
${\tan ^{ - 1}}\frac{{\sqrt 3 }}{5}$
- D
${\tan ^{ - 1}}\left( { - \frac{{\sqrt 3 }}{5}} \right)$
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For any complex number $w = c + id$, let $\arg ( w ) \in(-\pi, \pi]$, where $i =\sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z=x+$ iy satisfying arg $\left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4}$, the ordered pair $( x , y )$ lies on the circle
$x^2+y^2+5 x-3 y+4=0 .$
Then which of the following statements is (are) TRUE?
$(A)$ $\alpha=-1$ $(B)$ $\alpha \beta=4$ $(C)$ $\alpha \beta=-4$ $(D)$ $\beta=4$
For any complex number $w = c + id$, let $\arg ( w ) \in(-\pi, \pi]$, where $i =\sqrt{-1}$. Let $\alpha$ and $\beta$ be real numbers such that for all complex numbers $z=x+$ iy satisfying arg $\left(\frac{z+\alpha}{z+\beta}\right)=\frac{\pi}{4}$, the ordered pair $( x , y )$ lies on the circle
$x^2+y^2+5 x-3 y+4=0 .$
Then which of the following statements is (are) TRUE?
$(A)$ $\alpha=-1$ $(B)$ $\alpha \beta=4$ $(C)$ $\alpha \beta=-4$ $(D)$ $\beta=4$
- [IIT 2021]
Let $z$ be complex number satisfying $|z|^3+2 z^2+4 z-8=0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-$I$ to the correct entries in List-$II$.
List-$I$
List-$II$
($P$) $|z|^2$ is equal to
($1$) $12$
($Q$) $|z-\bar{z}|^2$ is equal to
($2$) $4$
($R$) $|z|^2+|z+\bar{z}|^2$ is equal to
($3$) $8$
($S$) $|z+1|^2$ is equal to
($4$) $10$
($5$) $7$
The correct option is:
Let $z$ be complex number satisfying $|z|^3+2 z^2+4 z-8=0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero.
Match each entry in List-$I$ to the correct entries in List-$II$.
| List-$I$ | List-$II$ |
| ($P$) $|z|^2$ is equal to | ($1$) $12$ |
| ($Q$) $|z-\bar{z}|^2$ is equal to | ($2$) $4$ |
| ($R$) $|z|^2+|z+\bar{z}|^2$ is equal to | ($3$) $8$ |
| ($S$) $|z+1|^2$ is equal to | ($4$) $10$ |
| ($5$) $7$ |
The correct option is:
- [IIT 2023]