If AB=4 and the ends A, B move on the coordinate axes, the locus of the mid-point of AB

Options

A. a straight line
B. a pair of straight lines
C. a circle
D. none of these

Answer

To find the locus of the midpoint of the segment AB, where the length AB = 4 and points A and B move along the coordinate axes, we can follow these steps:

### Step 1: Define Points A and B
Let the coordinates of point A on the x-axis be \( A(h, 0) \) and the coordinates of point B on the y-axis be \( B(0, k) \).

### Step 2: Use the Distance Formula
The distance \( AB \) can be calculated using the distance formula:
\[ AB = \sqrt{(h - 0)^2 + (0 - k)^2} = \sqrt{h^2 + k^2} \]
Given that \( AB = 4 \), we can set up the equation:
\[ \sqrt{h^2 + k^2} = 4 \]

### Step 3: Square Both Sides
To eliminate the square root, we square both sides of the equation:
\[ h^2 + k^2 = 16 \]

### Step 4: Rearranging the Equation
The equation \( h^2 + k^2 = 16 \) represents a circle with a radius of 4 centered at the origin (0, 0).

### Step 5: Conclusion
Thus, the locus of the midpoint M of segment AB, as points A and B move along the coordinate axes, is a circle with the equation:
\[ h^2 + k^2 = 16 \]