Find the coordinates of the points of the line segment joining (1, 3, 5) and (-2, 1, 2) in four
equal parts.
[Hint : Ratio may be 3 : 1, 1 : 1 or 1 : 3.]

Answer

To find the coordinates of the points that divide the line segment joining the points \( P(1, 3, 5) \) and \( Q(-2, 1, 2) \) into four equal parts, we will determine three points \( A \), \( B \), and \( C \) that lie on the segment. The ratios in which these points divide the segment are \( 1:3 \), \( 1:1 \), and \( 3:1 \) respectively.

### Step 1: Calculate the coordinates of point \( A \) (1:3 ratio)

Using the section formula, the coordinates of point \( A \) dividing the segment \( PQ \) in the ratio \( 1:3 \) can be calculated as follows:

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

Where:
- \( P(1, 3, 5) \) corresponds to \( (x_1, y_1, z_1) \)
- \( Q(-2, 1, 2) \) corresponds to \( (x_2, y_2, z_2) \)
- \( m = 1 \), \( n = 3 \)

Calculating the coordinates of \( A \):

\[ x_A = \frac{1 \cdot (-2) + 3 \cdot 1}{1 + 3} = \frac{-2 + 3}{4} = \frac{1}{4} \]

\[ y_A = \frac{1 \cdot 1 + 3 \cdot 3}{1 + 3} = \frac{1 + 9}{4} = \frac{10}{4} = \frac{5}{2} \]

\[ z_A = \frac{1 \cdot 2 + 3 \cdot 5}{1 + 3} = \frac{2 + 15}{4} = \frac{17}{4} \]

Thus, the coordinates of point \( A \) are:

\[ A\left( \frac{1}{4}, \frac{5}{2}, \frac{17}{4} \right) \]

### Step 2: Calculate the coordinates of point \( B \) (1:1 ratio)

Point \( B \) is the midpoint of segment \( PQ \). The midpoint formula gives us:

\[ B\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]

Calculating the coordinates of \( B \):

\[ x_B = \frac{1 + (-2)}{2} = \frac{-1}{2} \]

\[ y_B = \frac{3 + 1}{2} = \frac{4}{2} = 2 \]

\[ z_B = \frac{5 + 2}{2} = \frac{7}{2} \]

Thus, the coordinates of point \( B \) are:

\[ B\left( -\frac{1}{2}, 2, \frac{7}{2} \right) \]

### Step 3: Calculate the coordinates of point \( C \) (3:1 ratio)

Using the section formula again for point \( C \) dividing the segment \( PQ \) in the ratio \( 3:1 \):

\[ C\left( \frac{3 x_2 + 1 x_1}{3+1}, \frac{3 y_2 + 1 y_1}{3+1}, \frac{3 z_2 + 1 z_1}{3+1} \right) \]

Calculating the coordinates of \( C \):

\[ x_C = \frac{3 \cdot (-2) + 1 \cdot 1}{3 + 1} = \frac{-6 + 1}{4} = \frac{-5}{4} \]

\[ y_C = \frac{3 \cdot 1 + 1 \cdot 3}{3 + 1} = \frac{3 + 3}{4} = \frac{6}{4} = \frac{3}{2} \]

\[ z_C = \frac{3 \cdot 2 + 1 \cdot 5}{3 + 1} = \frac{6 + 5}{4} = \frac{11}{4} \]

Thus, the coordinates of point \( C \) are:

\[ C\left( -\frac{5}{4}, \frac{3}{2}, \frac{11}{4} \right) \]

### Final Result

The coordinates of the points dividing the line segment \( PQ \) into four equal parts are:

- \( A\left( \frac{1}{4}, \frac{5}{2}, \frac{17}{4} \right) \)
- \( B\left( -\frac{1}{2}, 2, \frac{7}{2} \right) \)
- \( C\left( -\frac{5}{4}, \frac{3}{2}, \frac{11}{4} \right) \)