A line intersects the y-axis and x-axis at the points P and Q respectively. If (2,-5) is the mid-point of PQ then find the coordinates of P and Q.

Answer

To find the coordinates of points P and Q where a line intersects the y-axis and x-axis respectively, given that (2, -5) is the midpoint of PQ, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the coordinates of points P and Q**:
- Since point P is on the y-axis, its coordinates can be represented as \( P(0, a) \) where \( a \) is the y-coordinate.
- Since point Q is on the x-axis, its coordinates can be represented as \( Q(b, 0) \) where \( b \) is the x-coordinate.

2. **Use the midpoint formula**:
- The midpoint \( M \) of a line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
- In our case, the midpoint \( M \) is given as \( (2, -5) \). Therefore, we can set up the equations:
\[ \frac{0 + b}{2} = 2 \quad \text{(for the x-coordinates)} \]
\[ \frac{a + 0}{2} = -5 \quad \text{(for the y-coordinates)} \]

3. **Solve for b**:
- From the equation \( \frac{0 + b}{2} = 2 \):
\[ b = 2 \times 2 = 4 \]

4. **Solve for a**:
- From the equation \( \frac{a + 0}{2} = -5 \):
\[ a = 2 \times -5 = -10 \]

5. **Write the coordinates of points P and Q**:
- Now that we have the values of \( a \) and \( b \):
- Point P is \( (0, a) = (0, -10) \)
- Point Q is \( (b, 0) = (4, 0) \)

### Final Answer:
- The coordinates of point P are \( (0, -10) \) and the coordinates of point Q are \( (4, 0) \).