Question

If x + iy = 1+2i /2+i
prove that x2 +
+ y2 =1.​

Answer 1

Question

If real numbers $x,y$ satisfy $x+yi=\dfrac{1+2i}{2+i}$, prove that $x^2+y^2=1$.

Solution

Let's find the real numbers.

$x+yi=\dfrac{1+2i}{2+i}$

$\implies x+yi=\dfrac{1+2i}{2+i} \times \dfrac{2-i}{2-i}$

$\implies x+yi=\dfrac{2-i+4i+2}{5}$

$\implies x+yi=\dfrac{4}{5}+\dfrac{3}{5} i$

It is worth noting that $x^2+y^2=(x)^2-(yi)^2=(x+yi)(x-yi)$.

$\implies x-yi=\dfrac{4}{5}-\dfrac{3}{5} i$

$\implies (x+yi)(x-yi)=(\dfrac{4}{5} +\dfrac{3}{5} i)(\dfrac{4}{5} -\dfrac{3}{5} i)$

$\implies x^2+y^2=\dfrac{16}{25} +\dfrac{9}{25} =\boxed{1}$