A, B and C are the points (a, p), (b,q) and (c,r) respectively such that a, b and c are in AP and p,q and r in GP. 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 solve the problem step-by-step, we need to analyze the conditions given for the points A, B, and C, and then apply the properties of arithmetic progression (AP) and geometric progression (GP) to find the relationship between p, q, and r.

### Step-by-Step Solution:

1. **Identify the Points**:
The points A, B, and C are given as:
- A = (a, p)
- B = (b, q)
- C = (c, r)

2. **Condition for AP**:
Since a, b, and c are in arithmetic progression (AP), we have the relationship:
\[ 2b = a + c \]

3. **Condition for GP**:
Since p, q, and r are in geometric progression (GP), we have the relationship:
\[ q^2 = pr \]

4. **Collinearity Condition**:
The points A, B, and C are collinear if the determinant formed by their coordinates is zero. The determinant can be expressed as:
\[ \begin{vmatrix} a & p & 1 \\ b & q & 1 \\ c & r & 1 \end{vmatrix} = 0 \]

5. **Expanding the Determinant**:
Expanding the determinant, we get:
\[ a(q - r) + b(r - p) + c(p - q) = 0 \]

6. **Substituting the AP Condition**:
Using the AP condition \(c = 2b - a\), we can substitute c in the determinant equation:
\[ a(q - r) + b(r - p) + (2b - a)(p - q) = 0 \]

7. **Simplifying the Equation**:
Expanding and simplifying the equation:
\[ a(q - r) + b(r - p) + 2b(p - q) - a(p - q) = 0 \]
Rearranging gives:
\[ a(q - r - p + q) + b(r - p + 2p - 2q) = 0 \]
This simplifies to:
\[ a(2q - p - r) + b(r - 2q + p) = 0 \]

8. **Setting Each Coefficient to Zero**:
For the equation to hold for arbitrary a and b, we must have:
\[ 2q - p - r = 0 \quad \text{(1)} \]
\[ r - 2q + p = 0 \quad \text{(2)} \]

9. **Solving the System of Equations**:
From equation (1), we can express r in terms of p and q:
\[ r = 2q - p \]
Substituting this into equation (2):
\[ (2q - p) - 2q + p = 0 \]
This simplifies to:
\[ 0 = 0 \]
This indicates that the equations are consistent.

10. **Conclusion**:
Since we have established that \(r = 2q - p\) and \(q^2 = pr\), we can deduce that:
\[ q^2 = p(2q - p) \]
This leads to the conclusion that \(p = q = r\), confirming that all three values are equal.

### Final Result:
Thus, if the points A, B, and C are collinear, then we conclude that:
\[ p = q = r \]