The point which divides the line segment joining the points (7, -6) and (3, 4) in ratio 1 : 2 internally lies in the

Options

A. I quadrant
B. II quadrant
C. III quadrant
D. IV quadrant

Answer

To solve the problem of finding the point that divides the line segment joining the points (7, -6) and (3, 4) in the ratio 1:2 internally, we will use the section formula.

### Step-by-Step Solution:

1. **Identify the Points and Ratio**:
Let the points be \( A(7, -6) \) and \( B(3, 4) \). The ratio in which the point divides the line segment is \( m:n = 1:2 \).

2. **Apply the Section Formula**:
The section formula states that if a point \( P(x, y) \) divides the line segment joining the points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( P \) can be calculated as:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

3. **Substitute the Values**:
Here, \( x_1 = 7 \), \( y_1 = -6 \), \( x_2 = 3 \), \( y_2 = 4 \), \( m = 1 \), and \( n = 2 \).
\[ P\left(\frac{1 \cdot 3 + 2 \cdot 7}{1 + 2}, \frac{1 \cdot 4 + 2 \cdot (-6)}{1 + 2}\right) \]

4. **Calculate the x-coordinate**:
\[ x = \frac{3 + 14}{3} = \frac{17}{3} \]

5. **Calculate the y-coordinate**:
\[ y = \frac{4 - 12}{3} = \frac{-8}{3} \]

6. **Final Coordinates**:
Thus, the coordinates of the point \( P \) that divides the line segment in the ratio 1:2 are:
\[ P\left(\frac{17}{3}, \frac{-8}{3}\right) \]

### Conclusion:
The point that divides the line segment joining the points (7, -6) and (3, 4) in the ratio 1:2 internally is \( P\left(\frac{17}{3}, \frac{-8}{3}\right) \).