If three points A,B and C have position vectors (1,x,3),(3,4,7) and (y,-2,-5), respectively and if they are collinear, then find (x,y).

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) such that the points A, B, and C with position vectors \( \mathbf{a} = (1, x, 3) \), \( \mathbf{b} = (3, 4, 7) \), and \( \mathbf{c} = (y, -2, -5) \) are collinear.

### Step-by-Step Solution:

1. **Understanding Collinearity**:
Points A, B, and C are collinear if the vectors \( \mathbf{AB} \) and \( \mathbf{AC} \) are parallel. This means there exists a scalar \( \lambda \) such that:
\[ \mathbf{AB} = \lambda \mathbf{AC} \]

2. **Finding Vectors**:
- The vector \( \mathbf{AB} \) is given by:
\[ \mathbf{AB} = \mathbf{b} - \mathbf{a} = (3, 4, 7) - (1, x, 3) = (3 - 1, 4 - x, 7 - 3) = (2, 4 - x, 4) \]
- The vector \( \mathbf{AC} \) is given by:
\[ \mathbf{AC} = \mathbf{c} - \mathbf{a} = (y, -2, -5) - (1, x, 3) = (y - 1, -2 - x, -5 - 3) = (y - 1, -2 - x, -8) \]

3. **Setting Up the Equation**:
From the collinearity condition, we have:
\[ (2, 4 - x, 4) = \lambda (y - 1, -2 - x, -8) \]

4. **Equating Components**:
This gives us three equations:
- From the first component:
\[ 2 = \lambda (y - 1) \quad \text{(1)} \]
- From the second component:
\[ 4 - x = \lambda (-2 - x) \quad \text{(2)} \]
- From the third component:
\[ 4 = \lambda (-8) \quad \text{(3)} \]

5. **Solving for \( \lambda \)**:
From equation (3):
\[ \lambda = -\frac{4}{8} = -\frac{1}{2} \]

6. **Substituting \( \lambda \) into Equation (1)**:
Substitute \( \lambda \) into equation (1):
\[ 2 = -\frac{1}{2}(y - 1) \]
Multiplying both sides by -2:
\[ -4 = y - 1 \implies y = -3 \]

7. **Substituting \( \lambda \) into Equation (2)**:
Substitute \( \lambda \) into equation (2):
\[ 4 - x = -\frac{1}{2}(-2 - x) \]
Simplifying:
\[ 4 - x = \frac{1}{2}(2 + x) \implies 4 - x = 1 + \frac{x}{2} \]
Multiplying through by 2 to eliminate the fraction:
\[ 8 - 2x = 2 + x \]
Rearranging gives:
\[ 8 - 2 = 2x \implies 6 = 3x \implies x = 2 \]

8. **Final Values**:
Thus, the values of \( x \) and \( y \) are:
\[ x = 2, \quad y = -3 \]

### Summary:
The final answer is:
\[ (x, y) = (2, -3) \]