If A=(2, -3, 7), B = (-1, 4, -5) and P is a point
on the line AB such that `AP : BP = 3 : 2` then P
has the coordinates

Options

A. `(4/5, -1/5, 11/5)`
B. `(1/5, 6/5, (-1)/5)`
C. `(7/5, (-18)/5, 29/5)`
D. `(-7, 18, -29)`

Answer

To find the coordinates of point P that divides the line segment AB in the ratio 3:2, we will use the section formula. We need to consider both the internal and external division of the segment.

### Step 1: Identify the coordinates of points A and B
Given:
- A = (2, -3, 7)
- B = (-1, 4, -5)

### Step 2: Use the internal section formula
The internal section formula for a point P that divides the line segment joining points A and B in the ratio m:n is given by:

\[ P = \left( \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n}, \frac{m \cdot z_2 + n \cdot z_1}{m+n} \right) \]

Here, we have:
- m = 3
- n = 2
- \( A(x_1, y_1, z_1) = (2, -3, 7) \)
- \( B(x_2, y_2, z_2) = (-1, 4, -5) \)

### Step 3: Substitute the values into the formula
Calculating the coordinates of P:

1. **X-coordinate**:
\[ P_x = \frac{3 \cdot (-1) + 2 \cdot 2}{3 + 2} = \frac{-3 + 4}{5} = \frac{1}{5} \]

2. **Y-coordinate**:
\[ P_y = \frac{3 \cdot 4 + 2 \cdot (-3)}{3 + 2} = \frac{12 - 6}{5} = \frac{6}{5} \]

3. **Z-coordinate**:
\[ P_z = \frac{3 \cdot (-5) + 2 \cdot 7}{3 + 2} = \frac{-15 + 14}{5} = \frac{-1}{5} \]

### Step 4: Combine the coordinates
Thus, the coordinates of point P when dividing AB internally in the ratio 3:2 are:

\[ P = \left( \frac{1}{5}, \frac{6}{5}, \frac{-1}{5} \right) \]

### Step 5: Use the external section formula
Next, we will find the coordinates of point P when dividing the segment externally in the same ratio.

The external section formula is given by:

\[ P = \left( \frac{m \cdot x_2 - n \cdot x_1}{m-n}, \frac{m \cdot y_2 - n \cdot y_1}{m-n}, \frac{m \cdot z_2 - n \cdot z_1}{m-n} \right) \]

### Step 6: Substitute the values into the external formula
Calculating the coordinates of P for external division:

1. **X-coordinate**:
\[ P_x = \frac{3 \cdot (-1) - 2 \cdot 2}{3 - 2} = \frac{-3 - 4}{1} = -7 \]

2. **Y-coordinate**:
\[ P_y = \frac{3 \cdot 4 - 2 \cdot (-3)}{3 - 2} = \frac{12 + 6}{1} = 18 \]

3. **Z-coordinate**:
\[ P_z = \frac{3 \cdot (-5) - 2 \cdot 7}{3 - 2} = \frac{-15 - 14}{1} = -29 \]

### Step 7: Combine the external coordinates
Thus, the coordinates of point P when dividing AB externally in the ratio 3:2 are:

\[ P = (-7, 18, -29) \]

### Summary of Results
- Internal division coordinates: \( P = \left( \frac{1}{5}, \frac{6}{5}, \frac{-1}{5} \right) \)
- External division coordinates: \( P = (-7, 18, -29) \)

### Final Answer
The coordinates of point P are \( \left( \frac{1}{5}, \frac{6}{5}, \frac{-1}{5} \right) \) for internal division and \( (-7, 18, -29) \) for external division.