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Highly selective Backlog Qs for JEE MainMathematicsComplex Number

If z ≠ 1 and z 2 z - 1 is real, then the point represented by the complex number z lies

Options

  1. AEither on the real axis or on a circle not passing through the origin.
  2. BOn the imaginary axis.
  3. CEither on the real axis or on a circle passing through the origin.
  4. DOn a circle with centre at the origin.

Correct answer

C. Either on the real axis or on a circle passing through the origin.

Step-by-step solution

z ≠ 1 , z 2 z - 1 is real If z is a real number, then z 2 z - 1 is real. Let z = x + iy ∴ x 2 - y 2 + 2 xiy x - 1 - iy x - 1 2 + y 2 equating imaginary part to zero. ⇒ - y x 2 - y 2 + 2 xy x - 1 = 0 ⇒ y x 2 + y 2 - 2 x = 0 ⇒ y = 0 or x 2 + y 2 - 2 x = 0 ∴ z lies on real axis or on a circle passing through origin.

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