If √a+ib = x+iy, then possible value of √a-ib is ​

Question

If \(\sqrt{a+ib}\) = x+iy, then possible value of\(\sqrt{a-ib}\) is 

A. x² + y²

B.  \(\sqrt{x^2 + y^2}\)

C. x + iy 

D. x – iy 

E. \(\sqrt{x^2 - y^2}\)

Answer 1

\(\sqrt{a+ib}\) = x+iy

Square both sides 

a + ib = (x + iy)² = x² + i2xy - y² 

So, we can say that a = x² – y² and b = 2xy 

a – ib = (x² – y²) – i(2xy) 

= (x)² + 2(x)(-iy) + (-iy)² 

= (x + (-iy))² 

= (x – iy)²

\(\sqrt{a-ib}\) = x-iy