If C and D are the points of internal and external division of line segment AB in the same ratio, then AC,AB, AD are in

Options

A. AP
B. GP
C. HP
D. AGP

Answer

To solve the problem, we need to show that the distances AC, AB, and AD are in Harmonic Progression (HP) when C and D divide the line segment AB in the same ratio, one internally and the other externally.

### Step-by-step Solution:

1. **Define Points A and B**:
Let point A be at the origin (0, 0) and point B be at (A, 0).

2. **Determine the Ratio**:
Let the ratio in which C divides AB internally and D divides AB externally be λ:1.

3. **Find Coordinates of C**:
Using the section formula for internal division, the coordinates of point C are given by:
\[ C = \left(\frac{A \cdot \lambda}{\lambda + 1}, 0\right) \]

4. **Find Coordinates of D**:
Using the section formula for external division, the coordinates of point D are given by:
\[ D = \left(\frac{A \cdot \lambda}{\lambda - 1}, 0\right) \]

5. **Calculate Distances**:
- Distance AC:
\[ AC = \frac{A \cdot \lambda}{\lambda + 1} - 0 = \frac{A \cdot \lambda}{\lambda + 1} \]
- Distance AB:
\[ AB = A - 0 = A \]
- Distance AD:
\[ AD = \frac{A \cdot \lambda}{\lambda - 1} - 0 = \frac{A \cdot \lambda}{\lambda - 1} \]

6. **Check for Harmonic Progression (HP)**:
For three quantities \(x\), \(y\), and \(z\) to be in HP, the following condition must hold:
\[ 2y = \frac{2xz}{x + z} \]
Here, let \(x = AC\), \(y = AB\), and \(z = AD\). We need to check if:
\[ 2A = \frac{2 \cdot \frac{A \cdot \lambda}{\lambda + 1} \cdot \frac{A \cdot \lambda}{\lambda - 1}}{\frac{A \cdot \lambda}{\lambda + 1} + \frac{A \cdot \lambda}{\lambda - 1}} \]

7. **Simplify the Right Side**:
- The numerator becomes:
\[ 2 \cdot \frac{A^2 \cdot \lambda^2}{(\lambda + 1)(\lambda - 1)} \]
- The denominator becomes:
\[ \frac{A \cdot \lambda \cdot (\lambda - 1) + A \cdot \lambda \cdot (\lambda + 1)}{(\lambda + 1)(\lambda - 1)} = \frac{A \cdot \lambda (2\lambda)}{(\lambda + 1)(\lambda - 1)} \]
- Thus, we have:
\[ 2A = \frac{2A^2 \cdot \lambda^2}{A \cdot \lambda \cdot 2\lambda} = A \]
- This confirms that the condition for HP holds true.

8. **Conclusion**:
Therefore, the distances AC, AB, and AD are in Harmonic Progression (HP).