Let z = x + iy be a non-zero complex number such that z² = i| z |², where i = √(- 1) , then z lies on the :

Solution

Maths sir · 4 steps

Expand z², then match real parts and imaginary parts separately.

1. Expand both sides. With z = x + iy, the left side is z² = (x² - y²) + 2xy i. The right side is i|z|² = 0 + (x² + y²) i, which is purely imaginary.
2. Match real parts. x² - y² = 0, so y = x or y = -x. This alone cannot decide between the two lines.
3. Match imaginary parts. 2xy = x² + y², which rearranges to a perfect square.

   x² - 2xy + y² = 0 ⇒ (x - y)² = 0 ⇒ y = x

4. So. Every non-zero z satisfying the condition has y = x, for example z = 1 + i gives z² = 2i = i|z|². So z lies on the line y = x, option (c).

The idea to keep. An equation between complex numbers is really two real equations. Write both, and use the second one to cut down whatever choices the first one leaves open.

Answer from the NTA final answer key. Solution by PrepWiser.