If |z - 25i| le 15 , then |max .amp(z)

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4-1.Complex numbers

hard

If $|z - 25i| \le 15$, then $|\max .amp(z) - \min .amp(z)| = $

${\cos ^{ - 1}}\left( {\frac{3}{5}} \right)$

$\pi - 2{\cos ^{ - 1}}\left( {\frac{3}{5}} \right)$

$\frac{\pi }{2} + {\cos ^{ - 1}}\left( {\frac{3}{5}} \right)$

${\sin ^{ - 1}}\left( {\frac{3}{5}} \right) - {\cos ^{ - 1}}\left( {\frac{3}{5}} \right)$

Solution

(b)We have
$max\ amp(z)=amp$$({z_2}),$ $min\ amp (z)=amp$$({z_1})$
Now $amp({z_1}) = {\theta _1} = {\cos ^{ – 1}}\left( {\frac{{15}}{{25}}} \right) = {\cos ^{ – 1}}\left( {\frac{3}{5}} \right)$
$amp({z_2}) = \frac{\pi }{2} + {\theta _2} = \frac{\pi }{2} + {\sin ^{ – 1}}\left( {\frac{{15}}{{25}}} \right) = \frac{\pi }{2} + {\sin ^{ – 1}}\left( {\frac{3}{5}} \right)$
$|\max \,\,amp(z) – \min \,\,amp(z)|$
$ = \left| {\frac{\pi }{2} + {{\sin }^{ – 1}}\frac{3}{5} – {{\cos }^{ – 1}}\frac{3}{5}} \right|$
$ = \left| {\frac{\pi }{2} + \frac{\pi }{2} – {{\cos }^{ – 1}}\frac{3}{5} – {{\cos }^{ – 1}}\frac{3}{5}} \right| = \pi – 2{\cos ^{ – 1}}\frac{3}{5}$

Standard 11

Mathematics

If $\alpha $ and $\beta $ are different complex numbers with $|\beta | = 1$, then $\left| {\frac{{\beta – \alpha }}{{1 – \overline \alpha \beta }}} \right|$ is equal to

medium

(IIT-1992)

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 ${z_1},{z_2}$ are two complex numbers such that $\left| {\frac{{{z_1} – {z_2}}}{{{z_1} + {z_2}}}} \right| = 1$ and $i{z_1} = k{z_2}$, where $k \in R$, then the angle between ${z_1} – {z_2}$ and ${z_1} + {z_2}$ is

hard

If $z_1 = 1+2i$ and $z_2 = 3+5i$ , then ${\mathop{\rm Re}\nolimits} \,\left( {\frac{{{{\overline Z }_2}{Z_1}}}{{{Z_2}}}} \right) = $

normal

If $z$ is a complex number such that $| z | = 4$ and $arg \,(z) = \frac {5\pi }{6}$ , then $z$ is equal to

normal

If $|z - 25i| \le 15$, then $|\max .amp(z) - \min .amp(z)| = $

Solution

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