If the equation of the locus of a point equidistant from the points `(a_1, b_1)` and `(a_2, b_2)` is `(a_1-a_2)x+(b_1-b_2)y+c=0` , then the value of `c` is `a a2-a2 2+b1 2-b2 2` `sqrt(a1 2+b1 2-a2 2-b2 2)` `1/2(a1 2+a2 2+b1 2+b2 2)` `1/2(a2 2+b2 2-a1 2-b1 2)`

Options

A. `a_(1)^(2)-a_(2)^(2)+b_(1)^(2)-b_(2)^(2)`
B. `sqrt((a_(1)^(2)+b_(1)^(2)-a_(2)^(2)-b_(2)^(2)))`
C. `1/2(a_(1)^(2)+a_(2)^(2)+b_(1)^(2)+b_(2)^(2))`
D. `1/2(a_(2)^(2)+b_(2)^(2)-a_(1)^(2)-b_(1)^(2))`

Answer

To find the value of \( c \) in the equation of the locus of a point equidistant from the points \( (a_1, b_1) \) and \( (a_2, b_2) \), we start with the condition that the distance from the point \( P(x, y) \) to both points is equal.

### Step 1: Set up the distance equations
The distances from point \( P(x, y) \) to points \( (a_1, b_1) \) and \( (a_2, b_2) \) are given by:
\[ PA = \sqrt{(x - a_1)^2 + (y - b_1)^2} \]
\[ PB = \sqrt{(x - a_2)^2 + (y - b_2)^2} \]

### Step 2: Set the distances equal
Since \( P \) is equidistant from both points, we have:
\[ \sqrt{(x - a_1)^2 + (y - b_1)^2} = \sqrt{(x - a_2)^2 + (y - b_2)^2} \]

### Step 3: Square both sides
Squaring both sides to eliminate the square roots gives:
\[ (x - a_1)^2 + (y - b_1)^2 = (x - a_2)^2 + (y - b_2)^2 \]

### Step 4: Expand both sides
Expanding both sides results in:
\[ (x^2 - 2a_1x + a_1^2 + y^2 - 2b_1y + b_1^2) = (x^2 - 2a_2x + a_2^2 + y^2 - 2b_2y + b_2^2) \]

### Step 5: Cancel common terms
The \( x^2 \) and \( y^2 \) terms cancel out:
\[ -2a_1x + a_1^2 - 2b_1y + b_1^2 = -2a_2x + a_2^2 - 2b_2y + b_2^2 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ -2a_1x + 2a_2x - 2b_1y + 2b_2y = a_2^2 - a_1^2 + b_2^2 - b_1^2 \]
This simplifies to:
\[ (2a_2 - 2a_1)x + (2b_2 - 2b_1)y = a_2^2 - a_1^2 + b_2^2 - b_1^2 \]

### Step 7: Divide by 2
Dividing the entire equation by 2 gives:
\[ (a_2 - a_1)x + (b_2 - b_1)y = \frac{1}{2}(a_2^2 - a_1^2 + b_2^2 - b_1^2) \]

### Step 8: Write in standard form
Rearranging this into the standard form \( (a_1 - a_2)x + (b_1 - b_2)y + c = 0 \), we identify:
\[ c = -\frac{1}{2}(a_2^2 - a_1^2 + b_2^2 - b_1^2) \]

### Step 9: Final expression for \( c \)
Thus, the value of \( c \) can be expressed as:
\[ c = \frac{1}{2}(b_1^2 + b_2^2 - a_1^2 - a_2^2) \]

### Conclusion
The correct option for \( c \) is:
\[ \frac{1}{2}(b_2^2 + a_2^2 - a_1^2 - b_1^2) \]