Find the co ordinates of the point which divides the line segment joining the points (6, 3) and (-4, 5) in the ratio 3 : 2 internally

Options

A. `(0,21/5)`
B. `(1,21/5)`
C. `(0,22/5)`
D. `(2,21/5)`

Explanation

To find the coordinates of the point that divides the line segment joining the points (6, 3) and (-4, 5) in the ratio 3:2 internally, we can use the section formula.

### Step-by-Step Solution:

1. **Identify the Points and the Ratio**:
- Let the points be \( A(6, 3) \) and \( B(-4, 5) \).
- The ratio in which the point divides the line segment is \( m:n = 3:2 \).

2. **Assign Coordinates**:
- Let \( A \) be \( (x_1, y_1) = (6, 3) \) and \( B \) be \( (x_2, y_2) = (-4, 5) \).
- Here, \( m = 3 \) and \( n = 2 \).

3. **Apply the Section Formula**:
The coordinates \( (x, y) \) of the point \( P \) that divides the line segment \( AB \) in the ratio \( m:n \) are given by the formula:
\[ x = \frac{m x_2 + n x_1}{m + n} \]
\[ y = \frac{m y_2 + n y_1}{m + n} \]

4. **Substitute the Values**:
- For \( x \):
\[ x = \frac{3 \cdot (-4) + 2 \cdot 6}{3 + 2} = \frac{-12 + 12}{5} = \frac{0}{5} = 0 \]
- For \( y \):
\[ y = \frac{3 \cdot 5 + 2 \cdot 3}{3 + 2} = \frac{15 + 6}{5} = \frac{21}{5} \]

5. **Final Coordinates**:
Thus, the coordinates of the point \( P \) that divides the segment \( AB \) in the ratio \( 3:2 \) are \( (0, \frac{21}{5}) \).

### Final Answer:
The coordinates of the point are \( \left( 0, \frac{21}{5} \right) \).