JEE PYQ: Complex Numbers Question 23
Question 23 - 2020 (06 Sep Shift 2)
Let $z = x + iy$ be a non-zero complex number such that $z^2 = i|z|^2$, where $i = \sqrt{-1}$, then $z$ lies on the:
(1) line, $y = -x$
(2) imaginary axis
(3) line, $y = x$
(4) real axis
Show Answer
Answer: (3)
Solution
$x^2 - y^2 + 2ixy = i(x^2+y^2)$. Real part: $x^2 - y^2 = 0 \Rightarrow y = \pm x$. Imaginary part: $2xy = x^2 + y^2$. With $y = x$: $2x^2 = 2x^2$ (true). With $y = -x$: $-2x^2 = 2x^2$ (false). So $z$ lies on $y = x$.
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