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 the coordinates of P and Q are, respectively.

Options

A. (0,-5) and (2,0)
B. (0,10) and (-4,0)
C. (0,4) and (-10, 0)
D. (0,-10) and (4,0)

Explanation

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 1: Define the Points
Let the coordinates of point P (where the line intersects the Y-axis) be (0, y1) and the coordinates of point Q (where the line intersects the X-axis) be (x2, 0).

### Step 2: Use the Midpoint Formula
The midpoint M of a line segment connecting two points (x1, y1) and (x2, y2) is given by:
\[ M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right) \]
In our case, the midpoint M is given as (2, -5). Therefore, we can set up the following equations:
\[ \frac{0 + x2}{2} = 2 \quad \text{(1)} \]
\[ \frac{y1 + 0}{2} = -5 \quad \text{(2)} \]

### Step 3: Solve for x2
From equation (1):
\[ \frac{x2}{2} = 2 \]
Multiplying both sides by 2 gives:
\[ x2 = 4 \]

### Step 4: Solve for y1
From equation (2):
\[ \frac{y1}{2} = -5 \]
Multiplying both sides by 2 gives:
\[ y1 = -10 \]

### Step 5: Write the Coordinates of P and Q
Now we can write the coordinates of points P and Q:
- Point P (0, y1) = (0, -10)
- Point Q (x2, 0) = (4, 0)

### Conclusion
Thus, the coordinates of points P and Q are:
- P = (0, -10)
- Q = (4, 0)