Let O be the origin and A be the point z _ 1 = 1 + 2i . If B is the...

Given $O$ be the origin and $A$ be the point ${z}_{1} = 1 + 2i$,

$B$ is the point ${z}_{2}, Re \left({z}_{2}\right) < 0$,

Also $OAB$ is a right angled isosceles triangle with $OB$ as hypotenuse,

Plotting the diagram we get,

Now by using concept of rotation we get,

$\dfrac{{z}_{2} - 0}{\left(1 + 2i\right) - 0} = \dfrac{\left|OB\right|}{\left|OA\right|}e\dfrac{i\pi}{4}$

$⇒ \dfrac{{z}_{2}}{1 + 2i} = \sqrt{2}e\dfrac{i\pi}{4}$

$Or{z}_{2} = \left(1 + 2i\right)\left(1 + i\right)$

$= - 1 + 3i$

Now finding $arg{z}_{2} = \pi - \tan^{- 1}\dfrac{\left|3\right|}{\left|1\right|} = \pi - \tan^{- 1}3$

Now \left|{z}_{2}\right| = \sqrt{{1}^{2} + {3}^{2}} = \sqrt{10}$</p><p>Now solving${z}_{1} - 2{z}_{2} = \left(1 + 2i\right) + 2 - 6i = 3 - 4i$</p><p>So,$arg\left({z}_{1} - 2{z}_{2}\right) = - \tan^{- 1}\dfrac{4}{3}$</p><p>Now solving \left|2{z}_{1} - {z}_{2}\right| = \left|2 + 4i + 1 - 3i\right| = \left|3 + i\right|$

$= \sqrt{10}$

So, \left|2{z}_{1} - {z}_{2}\right| = 5 is not true.