Let z¯ denote the complex conjugate of a complex number z and leti=−1 . In the set of complex numbers, the number of distinct roots of the equation z¯−z2=i(z¯+z2) is

Let z = x + iyx – iy – x2+ y2– 2ixy = i(x – iy + x2– y2) + 2ixy)(x – x2+ y2) – i(y + 2xy) = (y – 2xy) + i(x + x2– y2)⇒ x – x2+ y2= y – 2xy (1) x + x2– y2= –y – 2xy (2)(1) + (2)⇒ 2x = −4xy⇒x=−2xy⇒ x(1+2y)=0⇒ x=0 or y=−12Put x = 0 in (1) or (2) we gety2= y ⇒ y= 0, 1 ∴ 2 complex numbers are possible 0 + 0i and 0 + iput y=-12in (1) or (2) x-x2+14=−12+x⇒x2=34⇒x=±32 ∴32−i2 and −32−i2 are possible∴ 4 solutions are possible.

Let z¯ denote the complex conjugate of a complex number z and leti=−1 . In the set of complex numbers, the number of distinct roots of the equation z¯−z2=i(z¯+z2) is

Let z = x + iyx – iy – x2+ y2– 2ixy = i(x – iy + x2– y2) + 2ixy)(x – x2+ y2) – i(y + 2xy) = (y – 2xy) + i(x + x2– y2)⇒ x – x2+ y2= y – 2xy (1) x + x2– y2= –y – 2xy (2)(1) + (2)⇒ 2x = −4xy⇒x=−2xy⇒ x(1+2y)=0⇒ x=0 or y=−12Put x = 0 in (1) or (2) we gety2= y ⇒ y= 0, 1 ∴ 2 complex numbers are possible 0 + 0i and 0 + iput y=-12in (1) or (2) x-x2+14=−12+x⇒x2=34⇒x=±32 ∴32−i2 and −32−i2 are possible∴ 4 solutions are possible.

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Let $\bar{z}$ denote the complex conjugate of a complex number z and let $i = \sqrt{- 1}$ . In the set of complex numbers, the number of distinct roots of the equation $\bar{z} - z^{2} = i (\bar{z} + z^{2})$ is

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Detailed Solution

$\begin{array}{l} L e t z = x + i y \\ x - i y - x^{2} + y^{2} - 2 i x y = i (x - i y + x^{2} - y^{2}) + 2 i x y) \\ (x - x^{2} + y^{2}) - i (y + 2 x y) = (y - 2 x y) + i (x + x^{2} - y^{2}) \\ \Rightarrow x - x^{2} + y^{2} = y - 2 x y (1) \\ x + x^{2} - y^{2} = - y - 2 x y (2) \\ (1) + (2) ⇒ 2 x = - 4 x y \end{array}$

$\Rightarrow x = - 2 x y \Rightarrow x (1 + 2 y) = 0 \Rightarrow x = 0 o r y = - \frac{1}{2}$

$\begin{array}{l} P u t x = 0 i n (1) o r (2) w e g e t \\ y^{2} = y \Rightarrow y = 0, 1 \\ ∴ 2 c o m p l e x n u m b e r s a r e p o s s i b l e 0 + 0 i a n d 0 + i \end{array}$