Find the number of solutions of z2 + |z|2 = 0.

1.	Find the number of solutions of z2 + \|z\|2 = 0.
Answer» Given: ⇒ z²+\|z\|²=0 Let us assume z=x+iy ⇒ (x+iy)² + \(\sqrt{(x^2+y^2)^2}\) = 0 ⇒ x²+(iy)²+2(x)(iy)+x²+y²=0 ⇒ 2x²+y²+i²y²+i²xy=0 We know that i²=-1 ⇒ 2x²+y²-y²+i²xy=0 ⇒ 2x²+i²xy=0 Equating Real and Imaginary parts on both sides we get, ⇒ 2x²=0 and 2xy=0 ⇒ x=0 and y\(\varepsilon\)R ∴ z=0+iy where y\(\varepsilon\)R. i.e, Infinite solutions.

Answer»

Given:

⇒ z²+|z|²=0

Let us assume z=x+iy

⇒ (x+iy)² + \(\sqrt{(x^2+y^2)^2}\) = 0

⇒ x²+(iy)²+2(x)(iy)+x²+y²=0

⇒ 2x²+y²+i²y²+i²xy=0

We know that i²=-1

⇒ 2x²+y²-y²+i²xy=0

⇒ 2x²+i²xy=0

Equating Real and Imaginary parts on both sides we get,

⇒ 2x²=0 and 2xy=0

⇒ x=0 and y\(\varepsilon\)R

∴ z=0+iy where y\(\varepsilon\)R. i.e, Infinite solutions.

Discussion