If a point A lies on the line segment joining the points `P(6,0) and Q(0, 8)` such that `AP : AQ= 2 : 3`, find the co-ordinates of point A.

Answer

To find the coordinates of point A that divides the line segment joining points P(6, 0) and Q(0, 8) in the ratio AP:AQ = 2:3, we can use the formula for internal division of a line segment.

### Step-by-Step Solution:

1. **Identify the Coordinates of Points P and Q:**
- P = (6, 0)
- Q = (0, 8)

2. **Understand the Ratio:**
- The ratio AP:AQ = 2:3 means that point A divides the line segment PQ in the ratio of 2 parts from P to 3 parts from Q.

3. **Use the Internal Division Formula:**
- The coordinates of point A (x, y) that divides the line segment joining points (x1, y1) and (x2, y2) in the ratio m:n can be calculated using the formulas:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]
- Here, m = 2, n = 3, (x1, y1) = (6, 0), and (x2, y2) = (0, 8).

4. **Calculate the x-coordinate of A:**
\[ x = \frac{2 \cdot 0 + 3 \cdot 6}{2 + 3} = \frac{0 + 18}{5} = \frac{18}{5} \]

5. **Calculate the y-coordinate of A:**
\[ y = \frac{2 \cdot 8 + 3 \cdot 0}{2 + 3} = \frac{16 + 0}{5} = \frac{16}{5} \]

6. **Combine the Coordinates:**
- Therefore, the coordinates of point A are:
\[ A = \left(\frac{18}{5}, \frac{16}{5}\right) \]

### Final Answer:
The coordinates of point A are \(\left(\frac{18}{5}, \frac{16}{5}\right)\).

---