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 $$\bar{z}-z^{2}=i\left(\bar{z}+z^{2}\right)$$ is ____.

JEE Advanced · 2022 · 28 Aug 2022 · Paper 1 · Complex Numbers

Let

\overset{z}{ˉ}

denote the complex conjugate of a complex number

z

and let

i = - 1

. In the set of complex numbers, the number of distinct roots of the equation

\overset{z}{ˉ} - z^{2} = i (\overset{z}{ˉ} + z^{2})

is ____.

Answer: 4

From the given equation:

\overset{z}{ˉ} - z^{2} = i (\overset{z}{ˉ} + z^{2})

(1 - i) \overset{z}{ˉ} = (1 + i) z^{2}

\overset{z}{ˉ} = \frac{1 + i}{1 - i} z^{2} = \frac{( 1 + i ) ^{2}}{2} z^{2} = \frac{2 i}{2} z^{2} = i z^{2}

z^{2} = - i \overset{z}{ˉ}

.

Let

z = x + i y

. Then:

(x^{2} - y^{2}) + 2 i x y = - i (x - i y) = y - i x

Equating real and imaginary parts:

x^{2} - y^{2} = y and 2 x y = - x

From the second equation:

x (2 y + 1) = 0

, so

x = 0

y = - \frac{1}{2}

.

Case 1:

x = 0

: Then

- y^{2} = y

, so

y (y + 1) = 0

, giving

y = 0

y = 1

. Solutions:

(0, 0)

(0, 1)

.

Case 2:

y = - \frac{1}{2}

: Then

x^{2} - \frac{1}{4} = - \frac{1}{2}

, so

x^{2} = \frac{3}{4}

, giving

x = \pm \frac{3}{2}

. Solutions:

(\frac{3}{2}, - \frac{1}{2})

(- \frac{3}{2}, - \frac{1}{2})

.

Total: 4 distinct roots.

JEE Advanced · 2022 · 28 Aug 2022 · Paper 1 · Numerical answer