Find the coordinates of the point which divides the join of `A(-1, 7)` and `B(4, -3)` in the ratio `2:3.`

Options

A. `((10)/(3), (-20)/(3))`
B. `(2, (8)/(3))`
C. `(3,1)`
D. `(1, 3)`

Answer

To find the coordinates of the point that divides the line segment joining points A(-1, 7) and B(4, -3) 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 (x, y) can be calculated as follows:

\[ x = \frac{mx₂ + nx₁}{m+n} \]
\[ y = \frac{my₂ + ny₁}{m+n} \]

### Step-by-step Solution:

1. **Identify the coordinates of points A and B, and the ratio:**
- A(-1, 7) → (x₁, y₁) = (-1, 7)
- B(4, -3) → (x₂, y₂) = (4, -3)
- Ratio m:n = 2:3, where m = 2 and n = 3.

2. **Calculate the x-coordinate of point P:**
\[ x = \frac{mx₂ + nx₁}{m+n} = \frac{2 \cdot 4 + 3 \cdot (-1)}{2 + 3} \]
\[ = \frac{8 - 3}{5} = \frac{5}{5} = 1 \]

3. **Calculate the y-coordinate of point P:**
\[ y = \frac{my₂ + ny₁}{m+n} = \frac{2 \cdot (-3) + 3 \cdot 7}{2 + 3} \]
\[ = \frac{-6 + 21}{5} = \frac{15}{5} = 3 \]

4. **Combine the results to find the coordinates of point P:**
- The coordinates of point P are (1, 3).

### Final Answer:
The coordinates of the point which divides the join of A(-1, 7) and B(4, -3) in the ratio 2:3 are **(1, 3)**.