A straight line passes through the points `P(-1, 4)` and `Q(5,-2)`. It intersects x-axis at point A and y-axis at point B. M is the midpoint of the line segment AB. Find :
the co-ordinates of point M. 

Answer

To find the coordinates of point M, the midpoint of the line segment AB, we will follow these steps:

### Step 1: Find the slope of the line passing through points P and Q.
The slope (m) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For points \(P(-1, 4)\) and \(Q(5, -2)\):
- \(x_1 = -1\), \(y_1 = 4\)
- \(x_2 = 5\), \(y_2 = -2\)

Substituting the values:
\[ m = \frac{-2 - 4}{5 - (-1)} = \frac{-6}{6} = -1 \]

### Step 2: Use the slope-point form to find the equation of the line.
The slope-point form of the equation of a line is:
\[ y - y_1 = m(x - x_1) \]
Using point P(-1, 4) and the slope \(m = -1\):
\[ y - 4 = -1(x + 1) \]
Simplifying this:
\[ y - 4 = -x - 1 \implies y + x = 3 \]

### Step 3: Find the x-intercept (point A).
To find the x-intercept, set \(y = 0\) in the equation \(y + x = 3\):
\[ 0 + x = 3 \implies x = 3 \]
So, point A is \((3, 0)\).

### Step 4: Find the y-intercept (point B).
To find the y-intercept, set \(x = 0\) in the equation \(y + x = 3\):
\[ y + 0 = 3 \implies y = 3 \]
So, point B is \((0, 3)\).

### Step 5: Find the midpoint M of line segment AB.
The midpoint M of a line segment connecting points A \((x_1, y_1)\) and B \((x_2, y_2)\) is given by:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Substituting the coordinates of points A and B:
- A = (3, 0)
- B = (0, 3)

Calculating the midpoint:
\[ M = \left(\frac{3 + 0}{2}, \frac{0 + 3}{2}\right) = \left(\frac{3}{2}, \frac{3}{2}\right) \]

### Final Answer:
The coordinates of point M are \(\left(\frac{3}{2}, \frac{3}{2}\right)\).

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