Let OP.OQ=1 and let O,P and Q be three collinear points. If O and Q represent the complex numbers of origin and z respectively, then P represents

Options

A. `1/z`
B. `barz`
C. `1/barz`
D. `-z`

Explanation

To solve the problem, we need to find the complex number representation of point P given that O, P, and Q are collinear points and that OP * OQ = 1. Let's denote the complex numbers as follows:

- O (origin) = 0 (complex number)
- Q = z (complex number)
- P = z₁ (complex number we need to find)

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let O be the origin represented by the complex number 0.
- Let Q be represented by the complex number z.
- Let P be represented by the complex number z₁.

2. **Given Condition**:
- We know that OP * OQ = 1.
- Here, OP can be represented as |z₁| (the modulus of z₁) and OQ can be represented as |z| (the modulus of z).

3. **Expressing the Condition**:
- Therefore, we have |z₁| * |z| = 1.
- This implies that |z₁| = 1/|z|.

4. **Using the Argument**:
- Since O, P, and Q are collinear, the argument of z₁ (arg(z₁)) must be equal to arg(z) or arg(z) + π (because they can be in opposite directions).
- Let arg(z) = θ. Then, we have two cases for z₁:
- Case 1: z₁ = |z₁| e^(iθ)
- Case 2: z₁ = |z₁| e^(i(θ + π))

5. **Substituting the Modulus**:
- From |z₁| = 1/|z|, we can substitute this into our expressions for z₁:
- Case 1: z₁ = (1/|z|) e^(iθ)
- Case 2: z₁ = (1/|z|) e^(i(θ + π))

6. **Expressing in Terms of z**:
- Recall that z can be expressed in polar form as z = |z| e^(iθ).
- Therefore, the complex conjugate of z is given by z̅ = |z| e^(-iθ).
- Hence, we can express z₁ in terms of z:
- From Case 1: z₁ = 1/z̅
- From Case 2: z₁ = -1/z̅

7. **Final Representation**:
- Therefore, we conclude that P can be represented as:
- z₁ = 1/z̅ or z₁ = -1/z̅.

### Conclusion:
Thus, the complex number representing point P is either \( \frac{1}{\bar{z}} \) or \( -\frac{1}{\bar{z}} \).