A (-3,2) and B(5,4) are the end points of a lline segment ,find the co-ordinates of the point which divides AB in the ratio 2:3 :

Options

A. `(1/5 , (14)/(5))`
B. `(2,3)`
C. `(1/5 , 3)`
D. `(3,1/2)`

Answer

To find the coordinates of the point that divides the line segment AB in the ratio 2:3, we can use the section formula. The section formula states that if a point P divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of point P can be calculated using the following formulas:

\[ P_x = \frac{m \cdot x_2 + n \cdot x_1}{m + n} \]
\[ P_y = \frac{m \cdot y_2 + n \cdot y_1}{m + n} \]

### Step-by-Step Solution:

1. **Identify the coordinates of points A and B:**
- A = (-3, 2) → (x1, y1)
- B = (5, 4) → (x2, y2)

2. **Identify the ratio in which point P divides AB:**
- The ratio is 2:3, so m = 2 and n = 3.

3. **Substitute the values into the section formula for x-coordinate:**
\[ P_x = \frac{m \cdot x_2 + n \cdot x_1}{m + n} = \frac{2 \cdot 5 + 3 \cdot (-3)}{2 + 3} \]
\[ P_x = \frac{10 - 9}{5} = \frac{1}{5} \]

4. **Substitute the values into the section formula for y-coordinate:**
\[ P_y = \frac{m \cdot y_2 + n \cdot y_1}{m + n} = \frac{2 \cdot 4 + 3 \cdot 2}{2 + 3} \]
\[ P_y = \frac{8 + 6}{5} = \frac{14}{5} \]

5. **Combine the results to find the coordinates of point P:**
- Therefore, the coordinates of point P are:
\[ P = \left(\frac{1}{5}, \frac{14}{5}\right) \]

### Final Answer:
The coordinates of the point that divides the line segment AB in the ratio 2:3 are \( \left(\frac{1}{5}, \frac{14}{5}\right) \).