If `A-=(p,q,r)` and `B-=(p^('),q^('),r^('))` are two points on the line `lamdax=muy=vz` and `OA=a,OB=b` where `O` is origin then the value of `(pp^(')+qq^(')+rr^('))` is equal to (a)`a+b` (b)`ab` (c)`sqrt(a^2-b^2)` (d)`-ab`

Options

A. `a+b`
B. `ab`
C. `sqrt(a^(2)-b^(2))`
D. `-ab`

Explanation

To solve the problem, we need to find the value of \( pp' + qq' + rr' \) given the points \( A = (p, q, r) \) and \( B = (p', q', r') \) that lie on the line defined by \( \lambda x = \mu y = \nu z \) and the distances \( OA = a \) and \( OB = b \) from the origin \( O \).

### Step-by-Step Solution:

1. **Understanding the Line Equation**:
The line equation \( \lambda x = \mu y = \nu z \) indicates that the points \( A \) and \( B \) are collinear with the origin \( O \). This means that the points \( A \) and \( B \) can be expressed in terms of a parameter \( t \) as:
\[ A = k_1 (x_0, y_0, z_0) \quad \text{and} \quad B = k_2 (x_0, y_0, z_0) \]
where \( (x_0, y_0, z_0) \) is a direction vector along the line.

2. **Dot Product Representation**:
The dot product of the position vectors \( \vec{OA} \) and \( \vec{OB} \) can be expressed as:
\[ \vec{OA} \cdot \vec{OB} = |OA| |OB| \cos \theta \]
where \( |OA| = a \) and \( |OB| = b \).

3. **Angle Between Vectors**:
Since both points lie on the same line, the angle \( \theta \) between the vectors can either be \( 0 \) (if they point in the same direction) or \( \pi \) (if they point in opposite directions). Therefore, \( \cos \theta \) can be either \( 1 \) or \( -1 \).

4. **Calculating the Dot Product**:
Thus, we have:
\[ \vec{OA} \cdot \vec{OB} = ab \cos \theta \]
This leads to two possible results:
\[ \vec{OA} \cdot \vec{OB} = ab \quad \text{or} \quad \vec{OA} \cdot \vec{OB} = -ab \]

5. **Expressing the Dot Product**:
The dot product \( \vec{OA} \cdot \vec{OB} \) can also be expressed as:
\[ pp' + qq' + rr' \]
Therefore, we conclude:
\[ pp' + qq' + rr' = ab \quad \text{or} \quad pp' + qq' + rr' = -ab \]

6. **Final Answer**:
The value of \( pp' + qq' + rr' \) can be either \( ab \) or \( -ab \).

### Conclusion:
The correct options from the given choices are:
(b) \( ab \) and (d) \( -ab \).