Find the ratio in which the line-segment joining the points :
(i) (2,1,5) and (3,4,3) is divided by the plane :
x + y - z = ` (1)/(2)`
(ii) (1,2,3) and (-3,4,-5) is divided by the xy-plane

Answer

To solve the problem, we will find the ratio in which the line segments joining the given points are divided by the specified planes.

### Part (i): Points (2, 1, 5) and (3, 4, 3) divided by the plane \( x + y - z = \frac{1}{2} \)

1. **Identify the points and the plane equation**:
- Point 1: \( P_1(2, 1, 5) \)
- Point 2: \( P_2(3, 4, 3) \)
- Plane equation: \( x + y - z = \frac{1}{2} \)

2. **Convert the plane equation to standard form**:
- Multiply the entire equation by 2 to eliminate the fraction:
\[ 2x + 2y - 2z - 1 = 0 \]
This gives us \( a = 2, b = 2, c = -2, d = -1 \).

3. **Calculate \( P_1 \) and \( P_2 \)**:
- For \( P_1(2, 1, 5) \):
\[ P_1 = 2(2) + 2(1) - 2(5) - 1 = 4 + 2 - 10 - 1 = -5 \]
- For \( P_2(3, 4, 3) \):
\[ P_2 = 2(3) + 2(4) - 2(3) - 1 = 6 + 8 - 6 - 1 = 7 \]

4. **Calculate the ratio \( \lambda \)**:
- The ratio is given by:
\[ \lambda = -\frac{P_1}{P_2} = -\frac{-5}{7} = \frac{5}{7} \]

5. **Final ratio**:
- The ratio in which the line segment is divided is:
\[ \lambda : 1 = \frac{5}{7} : 1 = 5 : 7 \]

### Part (ii): Points (1, 2, 3) and (-3, 4, -5) divided by the xy-plane

1. **Identify the points and the plane equation**:
- Point 1: \( P_1(1, 2, 3) \)
- Point 2: \( P_2(-3, 4, -5) \)
- The xy-plane is given by \( z = 0 \).

2. **Calculate \( P_1 \) and \( P_2 \)**:
- For \( P_1(1, 2, 3) \):
\[ P_1 = 3 \]
- For \( P_2(-3, 4, -5) \):
\[ P_2 = -5 \]

3. **Calculate the ratio \( \lambda \)**:
- The ratio is given by:
\[ \lambda = -\frac{P_1}{P_2} = -\frac{3}{-5} = \frac{3}{5} \]

4. **Final ratio**:
- The ratio in which the line segment is divided is:
\[ \lambda : 1 = \frac{3}{5} : 1 = 3 : 5 \]

### Summary of Answers:
- (i) The ratio is \( 5 : 7 \).
- (ii) The ratio is \( 3 : 5 \).