Math, asked by shikshasangwan34, 2 months ago


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

Answers

Answered by user0888
17

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}

Similar questions