What are the coordinates of the point, which divides the join of the points (5,0) and (0,4) in the ratio 2 : 3 internally ?

Options

A. (8,-3)
B. (6,5)
C. `(3,(8)/(5))`
D. `(5/(2),2)`

Answer

To find the coordinates of the point that divides the line segment joining the points (5, 0) and (0, 4) in the ratio 2:3 internally, we can use the section formula.

### Step-by-step Solution:

1. **Identify the Points and Ratios**:
- Let the points be \( A(5, 0) \) and \( B(0, 4) \).
- The ratio in which the point divides the line segment is \( m_1:m_2 = 2:3 \).

2. **Assign Values**:
- From the points:
- \( x_1 = 5 \), \( y_1 = 0 \) (coordinates of point A)
- \( x_2 = 0 \), \( y_2 = 4 \) (coordinates of point B)
- From the ratio:
- \( m_1 = 2 \)
- \( m_2 = 3 \)

3. **Use the Section Formula**:
The coordinates \( (x, y) \) of the point that divides the line segment in the ratio \( m_1:m_2 \) are given by:
\[ x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2} \]
\[ y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \]

4. **Calculate the x-coordinate**:
\[ x = \frac{2 \cdot 0 + 3 \cdot 5}{2 + 3} = \frac{0 + 15}{5} = \frac{15}{5} = 3 \]

5. **Calculate the y-coordinate**:
\[ y = \frac{2 \cdot 4 + 3 \cdot 0}{2 + 3} = \frac{8 + 0}{5} = \frac{8}{5} \]

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

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