If $\frac{{z - i}}{{z + i}}(z \ne - i)$ is a purely imaginary number, then $z.\bar z$ is equal to
If $\frac{{z - i}}{{z + i}}(z \ne - i)$ is a purely imaginary number, then $z.\bar z$ is equal to
- A
$0$
- B
$1$
- C
$2$
- D
None of these
Similar Questions
Let $z$ be a complex number such that $\left| z \right| + z = 3 + i$ (where $i = \sqrt { - 1} $). Then $\left| z \right|$ is equal to
Let $z$ be a complex number such that $\left| z \right| + z = 3 + i$ (where $i = \sqrt { - 1} $). Then $\left| z \right|$ is equal to
- [JEE MAIN 2019]
If $z_1, z_2, z_3$ $\in$ $C$ such that $|z_1| = |z_2| = |z_3| = 2$, then greatest value of expression $|z_1 - z_2|.|z_2 - z_3| + |z_3 - z_1|.|z_1 - z_2| + |z_2 - z_3||z_3 - z_1|$ is
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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]
Let $Z$ and $W$ be complex numbers such that $\left| Z \right| = \left| W \right|,$ and arg $Z$ denotes the principal argument of $Z.$
Statement $1:$ If arg $Z+$ arg $W = \pi ,$ then $Z = -\overline W $.
Statement $2:$ $\left| Z \right| = \left| W \right|,$ implies arg $Z-$ arg $\overline W = \pi .$
Let $Z$ and $W$ be complex numbers such that $\left| Z \right| = \left| W \right|,$ and arg $Z$ denotes the principal argument of $Z.$
Statement $1:$ If arg $Z+$ arg $W = \pi ,$ then $Z = -\overline W $.
Statement $2:$ $\left| Z \right| = \left| W \right|,$ implies arg $Z-$ arg $\overline W = \pi .$
- [AIEEE 2012]