Find the co-ordinates of a point which divides the line joining the points (2, -1) and (3, 3) in the ratio 2 : 1 internally.

Answer

To find the coordinates of a point that divides the line segment joining the points (2, -1) and (3, 3) in the ratio 2:1 internally, we can use the section formula. The section formula for a point dividing a line segment internally in the ratio m:n is given by:

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

### Step-by-Step Solution:

1. **Identify the Points and the Ratio**:
- Let the points be \( A(2, -1) \) and \( B(3, 3) \).
- The ratio in which the point divides the line segment is \( m:n = 2:1 \).

2. **Assign Values**:
- From the points, we have:
- \( x_1 = 2 \), \( y_1 = -1 \)
- \( x_2 = 3 \), \( y_2 = 3 \)
- \( m = 2 \), \( n = 1 \)

3. **Apply the Section Formula**:
- Substitute the values into the section formula:
\[ x = \frac{mx_2 + nx_1}{m+n} = \frac{2 \cdot 3 + 1 \cdot 2}{2 + 1} \]
\[ y = \frac{my_2 + ny_1}{m+n} = \frac{2 \cdot 3 + 1 \cdot (-1)}{2 + 1} \]

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

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

6. **Final Coordinates**:
- Therefore, the coordinates of the point that divides the line segment in the ratio 2:1 are:
\[ \left( \frac{8}{3}, \frac{5}{3} \right) \]

### Final Answer:
The coordinates of the required point are \( \left( \frac{8}{3}, \frac{5}{3} \right) \).