Question

if X +i Y is equal to 2 + i / 2 - i then prove that X square + Y square is equal to 1​

Answer 1

The answer is this we use complex numbers to solve it

Answer 2

$x^{2} +y^{2} =1$, proved.

Step-by-step explanation:

We have,

x + iy = $\dfrac{2+i}{2-i}$

To prove that, $x^{2} +y^{2} =1.$

∴ x + iy = $\dfrac{2+i}{2-i}$

Rationalising numerator and denominator, we get

⇒ x + iy = $\dfrac{2+i}{2-i}\times \dfrac{2+i}{2+i}$

⇒ x + iy = $\dfrac{(2+i)^2}{2^2-i^2}$

⇒ x + iy = $\dfrac{4+4i+i^2}{4-(-1)}$

⇒ x + iy = $\dfrac{4+4i-1}{5}$ [ ∵ $i^{2} =-1$]

⇒ x + iy = $\dfrac{3}{5}+i\dfrac{4}{5}$

Comparing both sides, we get

∴ x = $\dfrac{3}{5}$ and y = $\dfrac{4}{5}$

L.H.S.= $\sqrt{x^{2}+y^{2}}$

= $\sqrt{(\dfrac{3}{5} )^{2}+(\dfrac{4}{5} )^{2}}$

$=\sqrt{\dfrac{9}{25} +\dfrac{16}{25}}$

$=\sqrt{\dfrac{9+16}{25}}$

$=\sqrt{\dfrac{25}{25}}$

= 1 = R.H.S., proved.