Find the coordinates of a point which divides externally the line joining (1, -3) and (-3, 9) in the ratio 1 : 3.

Answer

To find the coordinates of a point that divides the line segment joining the points (1, -3) and (-3, 9) externally in the ratio 1:3, we can use the formula for external division of a line segment.

### Step-by-step Solution:

1. **Identify the Points and Ratio**:
Let the points be \( A(1, -3) \) and \( B(-3, 9) \). The ratio in which the point divides the line externally is \( 1:3 \).

2. **Use the External Division Formula**:
The formula for the coordinates of a point \( P(x, y) \) that divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) externally in the ratio \( m:n \) is given by:
\[ P\left( \frac{mx_2 - nx_1}{m - n}, \frac{my_2 - ny_1}{m - n} \right) \]
Here, \( m = 1 \), \( n = 3 \), \( (x_1, y_1) = (1, -3) \), and \( (x_2, y_2) = (-3, 9) \).

3. **Substituting the Values**:
Substitute the values into the formula:
\[ x = \frac{1 \cdot (-3) - 3 \cdot 1}{1 - 3} = \frac{-3 - 3}{-2} = \frac{-6}{-2} = 3 \]
\[ y = \frac{1 \cdot 9 - 3 \cdot (-3)}{1 - 3} = \frac{9 + 9}{-2} = \frac{18}{-2} = -9 \]

4. **Final Coordinates**:
Therefore, the coordinates of the point that divides the line segment externally in the ratio 1:3 are \( (3, -9) \).

### Final Answer:
The required point is \( (3, -9) \).

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