OA and OB are two perpendicular straight lines. A straight line AB is drawn in such a manner that `OA+OB=8`. Find the locus of the mid point of AB.

Answer

To find the locus of the midpoint of line segment AB, where OA and OB are two perpendicular lines and OA + OB = 8, we can follow these steps:

### Step 1: Define the Coordinates
Let the point O be the origin (0, 0). Since OA and OB are perpendicular, we can assume:
- Point A lies on the x-axis, so its coordinates are (a, 0).
- Point B lies on the y-axis, so its coordinates are (0, b).

### Step 2: Express the Condition
According to the problem, the sum of OA and OB is given by:
\[ OA + OB = 8 \]
This translates to:
\[ a + b = 8 \]

### Step 3: Find the Midpoint of AB
The midpoint M of line segment AB can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of A and B:
\[ M = \left( \frac{a + 0}{2}, \frac{0 + b}{2} \right) = \left( \frac{a}{2}, \frac{b}{2} \right) \]

### Step 4: Express a and b in terms of x and y
Let the coordinates of the midpoint M be (x, y). Thus, we have:
\[ x = \frac{a}{2} \quad \text{and} \quad y = \frac{b}{2} \]
From these, we can express a and b as:
\[ a = 2x \quad \text{and} \quad b = 2y \]

### Step 5: Substitute into the Condition
Now substitute a and b into the equation \( a + b = 8 \):
\[ 2x + 2y = 8 \]
Dividing the entire equation by 2 gives:
\[ x + y = 4 \]

### Step 6: Conclusion
The locus of the midpoint M of line segment AB is the line given by the equation:
\[ x + y = 4 \]