Points $A(u)$, $B(v)$ and $C(z)$ in Argand diagram. Find $z$, given $BC=2AB$ and $\angle ABC=90^\circ$,$u=1+2{\sqrt{3}}i,\>v=3+2i$

Question

The complex number $u$ and $v$ are given by

$$u=1+2{\sqrt{3}}i\>\>\>\>\>v=3+2i$$

In the Argand diagram,$u$ and $v$ are represented by the points A and B. A third point lies in the first quadrant and is such that $BC=2AB$ and $\angle ABC=90^\circ$. Find the complex number $z$ represented by C.

Can anyone make me understand both the solutions and tell if there are any other ways to solve the question ?

Any help appreciated !!

Answer 1

Method 1:

Express the sides as,

$$BA = u - v,\>\>\>\>\>BC = z - v$$

Since $BC \perp BA$, we have $Arg(u-v) -Arg(z-v) = \frac\pi2$; and since $|BC| =2|BA|$, we have $|z-v| = 2|u-v|$. Together,

$$\frac{z-v}{u-v}=\frac{|z-v|}{|u-v|}e^{Arg(z-v) - Arg(u-v) }=2e^{-i\frac\pi2} = -2i$$

or,$$z = v -2i(u-v)$$

Solve with the givens $u=1+(2{\sqrt{3}})i$ and $v=3+2i$ to obtain,

$$z = (3+2i)-2i[-2-(2-2\sqrt3)]i= 4\sqrt3-1+6i$$

Method 2:

The second approach is actual similar to the first. Instead, it starts with

$$\frac{BC}{AB}=\frac{z-v}{v-u}=ke^{i\frac\pi2}=2i$$

The algebra afterwards is almost the same.