If complex numbers z₁ and z₂ both satisfy z + overline z = 2 | z -1

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4-1.Complex numbers

normal

If complex numbers $z_1$ and $z_2$ both satisfy $z + \overline z = 2 | z -1 |$ and $arg(z_1 -z_2) = \frac{\pi}{3} ,$ then value of $Im (z_1 + z_2)$ is, where $Im (z)$ denotes imaginary part of $z$ -

$\sin \frac{\pi }{3}$

$\cos ec \frac{\pi }{3}$

$\tan \frac{\pi }{3}$

$\cot \frac{\pi }{3}$

Solution

$\frac{z+\bar{z}}{2}=|z-1|$

$\Rightarrow \mathrm{x}^{2}=(\mathrm{x}-1)^{2}+\mathrm{y}^{2}$ or $\mathrm{y}^{2}=2 \mathrm{x}-1$

Let $\mathrm{z}_{1}\left(\frac{\mathrm{t}_{1}^{2}+1}{2}, \mathrm{t}_{1}\right) $ and $\mathrm{z}_{2}\left(\frac{\mathrm{t}_{2}^{2}+1}{2}, \mathrm{t}_{2}\right)$

$\because \arg \left(\mathrm{z}_{1}-\mathrm{z}_{2}\right)=\frac{\pi}{3} \Rightarrow \frac{\mathrm{t}_{2}-\mathrm{t}_{1}}{\frac{\mathrm{t}_{2}^{2}-\mathrm{t}_{1}}{2}}=\sqrt{3}$

$\Rightarrow \frac{\mathrm{2}}{\mathrm{t}_{1}+\mathrm{t}_{2}}=\sqrt{3}$

$\Rightarrow \operatorname{Im}\left(\mathrm{z}_{1}+\mathrm{z}_{2}\right)=\frac{2}{\sqrt{3}}$

Standard 11

Mathematics

If $z_1$ and $z_2$ are two unimodular complex numbers that satisfy $z_1^2 + z_2^2 = 5,$ then ${\left( {{z_1} – {{\bar z}_1}} \right)^2} + {\left( {{z_2} – {{\bar z}_2}} \right)^2}$ is equal to –

normal

${\left| {{z_1} + {z_2}} \right|^2} + {\left| {{z_1} – {z_2}} \right|^2}$ is equal to

hard

(AIEEE-2012)

If for $z=\alpha+i \beta,|z+2|=z+4(1+i)$, then $\alpha+\beta$ and $\alpha \beta$ are the roots of the equation

hard

(JEE MAIN-2023)

Let $z_1, z_2 \in C$ such that $| z_1 + z_2 |= \sqrt 3$ and $|z_1| = |z_2| = 1,$ then the value of $|z_1 – z_2|$ is

normal

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z^2+z+1\right|=1$. Then which of the following statements is/are $TRUE$?

$(A)$ $\left|z+\frac{1}{2}\right| \leq \frac{1}{2}$ for all $z \in S$ $(B)$ $|z| \leq 2$ for all $z \in S$

$(C)$ $\left|z+\frac{1}{2}\right| \geq \frac{1}{2}$ for all $z \in S$ $(D)$ The set $S$ has exactly four elements

normal

(IIT-2020)

If complex numbers $z_1$ and $z_2$ both satisfy $z + \overline z = 2 | z -1 |$ and $arg(z_1 -z_2) = \frac{\pi}{3} ,$ then value of $Im (z_1 + z_2)$ is, where $Im (z)$ denotes imaginary part of $z$ -

Solution

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If $z_1$ and $z_2$ are two unimodular complex numbers that satisfy $z_1^2 + z_2^2 = 5,$ then ${\left( {{z_1} – {{\bar z}_1}} \right)^2} + {\left( {{z_2} – {{\bar z}_2}} \right)^2}$ is equal to –

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