If `z!=1` and `(z^2)/(z-1)` is real, then the point represented by the complex number z lies (1) either on the real axis or on a circle passing through the origin (2) on a circle with centre at the origin (3) either on the real axis or on a circle not passing through the origin (4) on the imaginary axis

Options

A. either on the real axis or on a circle passing thorugh the origin.
B. on a circle with centre at the origin.
C. either on the real axis or an a circle not possing through the origin .
D. on the imaginary axis .

Explanation

To solve the problem, we need to analyze the condition under which the expression \(\frac{z^2}{z-1}\) is real. Let's denote \(z\) as a complex number \(z = x + iy\), where \(x\) and \(y\) are real numbers.

### Step-by-Step Solution:

1. **Substituting \(z\)**:
\[ z = x + iy \]
Then,
\[ z^2 = (x + iy)^2 = x^2 - y^2 + 2xyi \]
and
\[ z - 1 = (x - 1) + iy. \]

2. **Forming the Expression**:
We can write the expression \(\frac{z^2}{z-1}\):
\[ \frac{z^2}{z-1} = \frac{x^2 - y^2 + 2xyi}{(x - 1) + iy}. \]

3. **Rationalizing the Denominator**:
To rationalize, we multiply the numerator and the denominator by the conjugate of the denominator:
\[ \frac{(x^2 - y^2 + 2xyi)((x - 1) - iy)}{((x - 1) + iy)((x - 1) - iy)}. \]
The denominator simplifies to:
\[ (x - 1)^2 + y^2. \]

4. **Calculating the Numerator**:
The numerator becomes:
\[ (x^2 - y^2)(x - 1) - (x^2 - y^2)iy + 2xyi(x - 1) - 2xyy. \]
Simplifying this, we can separate the real and imaginary parts.

5. **Setting the Imaginary Part to Zero**:
For the expression to be real, the imaginary part must equal zero. This leads to:
\[ 2xy(x - 1) - (x^2 - y^2)y = 0. \]
Factoring out \(y\):
\[ y(2x(x - 1) - (x^2 - y^2)) = 0. \]

6. **Finding Solutions**:
This gives us two cases:
- Case 1: \(y = 0\) (This corresponds to points on the real axis).
- Case 2: \(2x(x - 1) - (x^2 - y^2) = 0\).

7. **Analyzing the Second Case**:
Rearranging the second case gives:
\[ x^2 + y^2 - 2x = 0, \]
which can be rewritten as:
\[ (x - 1)^2 + y^2 = 1. \]
This represents a circle with center at \((1, 0)\) and radius 1.

### Conclusion:
The points represented by the complex number \(z\) either lie on the real axis (when \(y = 0\)) or on the circle defined by \((x - 1)^2 + y^2 = 1\).

Thus, the correct option is:
**(1) either on the real axis or on a circle passing through the origin.**