The point which divides the line segment joining the points (8,–9) and (2,3) in ratio 1 : 2 internally lies in the

Options

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

Explanation

To solve the problem of finding the point that divides the line segment joining the points (8, -9) and (2, 3) in the ratio 1:2 internally, we will use the section formula. Let's go through the steps:

### Step-by-Step Solution:

1. **Identify the Points and the Ratio:**
- Let point A = (8, -9) and point B = (2, 3).
- The ratio in which point P divides the line segment AB is 1:2.

2. **Assign Coordinates:**
- Let the coordinates of point A be (x1, y1) = (8, -9).
- Let the coordinates of point B be (x2, y2) = (2, 3).
- The ratio is m:n = 1:2, where m = 1 and n = 2.

3. **Use the Section Formula:**
- The coordinates of point P (x, y) that divides the line segment AB in the ratio m:n are given by:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

4. **Substituting the Values:**
- For x-coordinate:
\[ x = \frac{1 \cdot 2 + 2 \cdot 8}{1 + 2} = \frac{2 + 16}{3} = \frac{18}{3} = 6 \]
- For y-coordinate:
\[ y = \frac{1 \cdot 3 + 2 \cdot (-9)}{1 + 2} = \frac{3 - 18}{3} = \frac{-15}{3} = -5 \]

5. **Coordinates of Point P:**
- Therefore, the coordinates of point P are (6, -5).

6. **Determine the Quadrant:**
- The x-coordinate is positive (6) and the y-coordinate is negative (-5).
- This means point P lies in the fourth quadrant.

### Final Answer:
The point which divides the line segment joining the points (8, -9) and (2, 3) in the ratio 1:2 internally lies in the **fourth quadrant**.