Find the coordinates of the points which divide the line joining the points (2,-4,3), (-4,5,-6) in the ratio (i) 1: 4 (ii) 2:1.

Answer

To find the coordinates of the points that divide the line segment joining the points \( A(2, -4, 3) \) and \( B(-4, 5, -6) \) in the given ratios, we can use the section formula. The section formula states that if a point \( P(x, y, z) \) divides the line segment joining the points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) in the ratio \( m:n \), then the coordinates of point \( P \) are given by:

\[ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]

### (i) For the ratio \( 1:4 \)

1. **Identify the coordinates and the ratio:**
- \( A(2, -4, 3) \) and \( B(-4, 5, -6) \)
- Ratio \( m:n = 1:4 \) (where \( m = 1 \) and \( n = 4 \))

2. **Calculate the x-coordinate:**
\[ x = \frac{(-4) \cdot 1 + 2 \cdot 4}{1 + 4} = \frac{-4 + 8}{5} = \frac{4}{5} \]

3. **Calculate the y-coordinate:**
\[ y = \frac{5 \cdot 1 + (-4) \cdot 4}{1 + 4} = \frac{5 - 16}{5} = \frac{-11}{5} \]

4. **Calculate the z-coordinate:**
\[ z = \frac{(-6) \cdot 1 + 3 \cdot 4}{1 + 4} = \frac{-6 + 12}{5} = \frac{6}{5} \]

5. **Final coordinates of point Q:**
\[ Q\left( \frac{4}{5}, \frac{-11}{5}, \frac{6}{5} \right) \]

### (ii) For the ratio \( 2:1 \)

1. **Identify the coordinates and the ratio:**
- \( A(2, -4, 3) \) and \( B(-4, 5, -6) \)
- Ratio \( m:n = 2:1 \) (where \( m = 2 \) and \( n = 1 \))

2. **Calculate the x-coordinate:**
\[ x = \frac{(-4) \cdot 2 + 2 \cdot 1}{2 + 1} = \frac{-8 + 2}{3} = \frac{-6}{3} = -2 \]

3. **Calculate the y-coordinate:**
\[ y = \frac{5 \cdot 2 + (-4) \cdot 1}{2 + 1} = \frac{10 - 4}{3} = \frac{6}{3} = 2 \]

4. **Calculate the z-coordinate:**
\[ z = \frac{(-6) \cdot 2 + 3 \cdot 1}{2 + 1} = \frac{-12 + 3}{3} = \frac{-9}{3} = -3 \]

5. **Final coordinates of point P:**
\[ P(-2, 2, -3) \]

### Summary of Results:
- Coordinates of point \( Q \) (dividing in ratio \( 1:4 \)): \( \left( \frac{4}{5}, \frac{-11}{5}, \frac{6}{5} \right) \)
- Coordinates of point \( P \) (dividing in ratio \( 2:1 \)): \( (-2, 2, -3) \)