A,B,C are the points (a,p),( b,q) and (c,r) respectively such that a,b,c are in A.P. and p,q,r in G.P. If the points are collinear then

Options

A. `p=q=r`
B. `p^(2)=q`
C. `q^(2)=r`
D. `r^(2)=p`

Answer

To determine the conditions under which the points A, B, and C are collinear, given that A, B, C are the points (a, p), (b, q), and (c, r) respectively, where a, b, c are in Arithmetic Progression (A.P.) and p, q, r are in Geometric Progression (G.P.), we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Conditions**:
- Points A, B, and C are given as (a, p), (b, q), and (c, r).
- The points A, B, C are collinear if the area of the triangle formed by these points is zero. This can be determined using the determinant method.

2. **Setting Up the Determinant**:
- The area of the triangle formed by the points A, B, and C can be expressed using the determinant:
\[ \Delta = \begin{vmatrix} a & p & 1 \\ b & q & 1 \\ c & r & 1 \end{vmatrix} \]
- For the points to be collinear, we need \(\Delta = 0\).

3. **Calculating the Determinant**:
- Expanding the determinant, we have:
\[ \Delta = a(q - r) + b(r - p) + c(p - q) \]
- Setting this equal to zero gives us:
\[ a(q - r) + b(r - p) + c(p - q) = 0 \]

4. **Using the A.P. Condition**:
- Since a, b, c are in A.P., we have:
\[ 2b = a + c \quad \text{(1)} \]

5. **Using the G.P. Condition**:
- Since p, q, r are in G.P., we have:
\[ q^2 = pr \quad \text{(2)} \]

6. **Substituting A.P. into the Determinant**:
- Substitute \(a + c = 2b\) into the determinant equation:
\[ 2b(q - r) + b(r - p) + c(p - q) = 0 \]
- Rearranging gives:
\[ 2b(q - r) + b(r - p) + (2b - a)(p - q) = 0 \]

7. **Simplifying the Equation**:
- This leads to:
\[ 2b(q - r) + b(r - p) + 2b(p - q) - a(p - q) = 0 \]
- Collecting terms leads to a relationship between p, q, and r.

8. **Final Relationship**:
- From the conditions derived, we conclude that:
\[ p + r = 2q \]
- This indicates that p, q, and r must be equal for both the A.P. and G.P. conditions to hold true.

### Conclusion
Thus, for the points A, B, and C to be collinear, it must be that \(p = q = r\).