A line moves in such a way that the sum of the intercepts made by it on the axes is always c. The locus of the mid- point of its intercept between the axes is (A) `x+y =2c` (B) `x+y=c` (C) `2(x+y)=c` (D) None of these

Options

A. x+y=2c
B. x+y=c
C. 2(x+y)=c
D. 2x+y=c

Explanation

To find the locus of the midpoint of the intercepts made by a line on the axes, given that the sum of the intercepts is always a constant \( c \), we can follow these steps:

### Step 1: Understand the intercepts
Let the x-intercept of the line be \( a \) and the y-intercept be \( b \). According to the problem, the sum of the intercepts is given by:
\[ a + b = c \]

### Step 2: Express the intercepts in terms of the midpoint
The midpoint \( M \) of the intercepts \( (a, 0) \) and \( (0, b) \) can be calculated as:
\[ M\left(\frac{a + 0}{2}, \frac{0 + b}{2}\right) = \left(\frac{a}{2}, \frac{b}{2}\right) \]
Let \( x = \frac{a}{2} \) and \( y = \frac{b}{2} \). Therefore, we can express \( a \) and \( b \) in terms of \( x \) and \( y \):
\[ a = 2x \quad \text{and} \quad b = 2y \]

### Step 3: Substitute into the intercept sum equation
Substituting \( a \) and \( b \) into the intercept sum equation:
\[ 2x + 2y = c \]

### Step 4: Simplify the equation
Dividing the entire equation by 2 gives:
\[ x + y = \frac{c}{2} \]

### Step 5: Determine the locus
The equation \( x + y = \frac{c}{2} \) represents a straight line in the coordinate plane.

### Conclusion
Thus, the locus of the midpoint of the intercepts is:
\[ 2(x + y) = c \]
This corresponds to option (C).