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  4. On the Argand plane point ' A ' denotes a complex number z 1. A triangle OBQ is made directily similiar to the triangle OAM, where OM =1 as shown in the figure. If the point B denotes the complex number z2, then the complex number corresponding to the point ' Q ' is

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Q. On the Argand plane point ' $A$ ' denotes a complex number $z _1$. A triangle $OBQ$ is made directily similiar to the triangle $OAM$, where $OM =1$ as shown in the figure. If the point $B$ denotes the complex number $z_2$, then the complex number corresponding to the point ' $Q$ ' isMathematics Question Image

Complex Numbers and Quadratic Equations

Solution:

$\frac{z_2}{\left|z_2\right|}=\frac{z}{|z|} e^{i \theta}$.....(1)
$\frac{z_1}{\left|z_1\right|}=1 e^{i \theta}$....(2)
substitute the value of $e ^{ i \theta}$ from (2) in (1)
$\frac{ z }{| z |}=\frac{ z _2}{\left| z _2\right|} \cdot \frac{\left| z _1\right|}{ z _1} \Rightarrow \frac{ z }{| z |}=\frac{ z _2 / z _1}{\left| z _2 / z _1\right|} ; z =\frac{ z _2}{ z _1} $