A point C divides the line AC, where A(1, 3) and B(2, 7) in the ratio of `3:4` . The coordinates of C are

Options

A. `(5/3, 5)`
B. `(-2, -9)`
C. `(3/5, 5)`
D. `(10/7 , 33/7)`

Answer

To find the coordinates of point C that divides the line segment AB in the ratio 3:4, we can use the section formula. The coordinates of points A and B are given as A(1, 3) and B(2, 7).

### Step-by-Step Solution:

1. **Identify the coordinates of points A and B:**
- A = (x1, y1) = (1, 3)
- B = (x2, y2) = (2, 7)

2. **Identify the ratio in which point C divides the line segment AB:**
- The ratio is given as m:n = 3:4, where m = 3 and n = 4.

3. **Use the section formula to find the coordinates of point C:**
- The section formula for the coordinates of point C (x, y) that divides the line segment joining points A and B in the ratio m:n is given by:
\[ C\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

4. **Substitute the values into the formula:**
- For the x-coordinate:
\[ x = \frac{3 \cdot 2 + 4 \cdot 1}{3 + 4} = \frac{6 + 4}{7} = \frac{10}{7} \]
- For the y-coordinate:
\[ y = \frac{3 \cdot 7 + 4 \cdot 3}{3 + 4} = \frac{21 + 12}{7} = \frac{33}{7} \]

5. **Combine the results to find the coordinates of point C:**
- Therefore, the coordinates of point C are:
\[ C\left( \frac{10}{7}, \frac{33}{7} \right) \]

### Final Answer:
The coordinates of point C are \( C\left( \frac{10}{7}, \frac{33}{7} \right) \).

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