To solve the problem, we need to analyze the statements given regarding the line segment AB, which is divided internally at point P and externally at point Q in the same ratio, with M as the midpoint of AB.
### Step 1: Define the coordinates of points A and B
Let's place point A at the origin (0, 0) and point B at (10, 0).
### Step 2: Find the coordinates of point P
Since P divides AB internally in the ratio k:1, we can use the section formula. The coordinates of P can be calculated as:
\[
P = \left( \frac{k \cdot x_B + 1 \cdot x_A}{k + 1}, \frac{k \cdot y_B + 1 \cdot y_A}{k + 1} \right)
\]
Substituting the coordinates of A and B:
\[
P = \left( \frac{k \cdot 10 + 1 \cdot 0}{k + 1}, \frac{k \cdot 0 + 1 \cdot 0}{k + 1} \right) = \left( \frac{10k}{k + 1}, 0 \right)
\]
### Step 3: Find the coordinates of point Q
Point Q divides AB externally in the same ratio k:1. The coordinates of Q can be calculated as:
\[
Q = \left( \frac{k \cdot x_B - 1 \cdot x_A}{k - 1}, \frac{k \cdot y_B - 1 \cdot y_A}{k - 1} \right)
\]
Substituting the coordinates of A and B:
\[
Q = \left( \frac{k \cdot 10 - 1 \cdot 0}{k - 1}, \frac{k \cdot 0 - 1 \cdot 0}{k - 1} \right) = \left( \frac{10k}{k - 1}, 0 \right)
\]
### Step 4: Find the coordinates of midpoint M
The midpoint M of segment AB is given by:
\[
M = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) = \left( \frac{0 + 10}{2}, \frac{0 + 0}{2} \right) = (5, 0)
\]
### Step 5: Calculate distances MP, MB, and MQ
Using the distance formula:
- \( MP = |x_P - x_M| = \left| \frac{10k}{k + 1} - 5 \right| \)
- \( MB = |x_B - x_M| = |10 - 5| = 5 \)
- \( MQ = |x_Q - x_M| = \left| \frac{10k}{k - 1} - 5 \right| \)
### Step 6: Check if MP, MB, and MQ are in G.P.
For MP, MB, and MQ to be in G.P., the following condition must hold:
\[
MP^2 = MB \cdot MQ
\]
Substituting the distances calculated in Step 5 into this equation will help us determine if they are in G.P.
### Step 7: Calculate AP, AB, and AQ
- \( AP = |x_P - x_A| = \left| \frac{10k}{k + 1} - 0 \right| = \frac{10k}{k + 1} \)
- \( AB = |x_B - x_A| = |10 - 0| = 10 \)
- \( AQ = |x_Q - x_A| = \left| \frac{10k}{k - 1} - 0 \right| = \frac{10k}{k - 1} \)
### Step 8: Check if AP, AB, and AQ are in H.P.
For AP, AB, and AQ to be in H.P., the reciprocals must be in A.P.:
\[
\frac{1}{AP}, \frac{1}{AB}, \frac{1}{AQ}
\]
This means checking if:
\[
\frac{1}{AP} - \frac{1}{AB} = \frac{1}{AQ} - \frac{1}{AP}
\]
### Conclusion
After performing the calculations, we can conclude whether the statements are true or false based on the conditions derived from the distances.
