Question

If z ≠ 1 and `"z"^2/("z - 1")` is real, then the point represented by the complex number z lies ______.

Options

• either on the real axis or on a circle passing through the origin

• on a circle with centre at the origin

• either on the real axis or on a circle not passing through the origin

• on the imaginary axis

Answer 1

If z ≠ 1 and `"z"^2/("z - 1")` is real, then the point represented by the complex number z lies either on the real axis or on a circle passing through the origin.

Explanation:

Let z = x + iy

Then, z² = (x² - y²) + i (2xy)

`therefore "z"^2/("z - 1") = ((x^2 - y^2) + "i"(2xy))/(x + "iy" - 1)`

`= ((x^2 - y^2) + "i"(2xy))/(x - 1 + "i"y) xx (x - 1 - "i"y)/(x - 1 - "iy")`

Since, `"z"^2/("z - 1")` is real.

∴ its imaginary part = 0

⇒ 2xy(x - 1) - y(x² - y²) = 0

⇒ y(x² - 2x + y²) = 0

⇒ y = 0 or x² - 2x + y² = 0

∴ z lies either on real axis or on a circle passing through origin.