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_2+b_2)y+c=0`, then the value of C is

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_2^2+b_2^2-a_2^2-b_1^2)`
D. `(1)/(2)(a_1^2+b_2^2+a_1^2+b_2^2)`

Explanation

To find the value of \( c \) in the equation of the locus of a point that is equidistant from the points \( (a_1, b_1) \) and \( (a_2, b_2) \), we can follow these steps:

### Step 1: Set Up the Distance Equation
Let the point \( P(h, k) \) be equidistant from the points \( A(a_1, b_1) \) and \( B(a_2, b_2) \). According to the distance formula, we have:
\[ PA = PB \]
This translates to:
\[ \sqrt{(h - a_1)^2 + (k - b_1)^2} = \sqrt{(h - a_2)^2 + (k - b_2)^2} \]

### Step 2: Square Both Sides
To eliminate the square roots, we square both sides:
\[ (h - a_1)^2 + (k - b_1)^2 = (h - a_2)^2 + (k - b_2)^2 \]

### Step 3: Expand Both Sides
Now, we expand both sides:
\[ (h^2 - 2ha_1 + a_1^2 + k^2 - 2kb_1 + b_1^2) = (h^2 - 2ha_2 + a_2^2 + k^2 - 2kb_2 + b_2^2) \]

### Step 4: Simplify the Equation
Cancel \( h^2 \) and \( k^2 \) from both sides:
\[ -2ha_1 - 2kb_1 + a_1^2 + b_1^2 = -2ha_2 - 2kb_2 + a_2^2 + b_2^2 \]
Rearranging gives:
\[ -2ha_1 + 2ha_2 - 2kb_1 + 2kb_2 = a_2^2 + b_2^2 - a_1^2 - b_1^2 \]

### Step 5: Factor Out Common Terms
Factor out \( 2h \) and \( 2k \):
\[ 2h(a_2 - a_1) + 2k(b_2 - b_1) = a_2^2 + b_2^2 - a_1^2 - b_1^2 \]
Dividing through by 2 gives:
\[ h(a_2 - a_1) + k(b_2 - b_1) = \frac{1}{2}(a_2^2 + b_2^2 - a_1^2 - b_1^2) \]

### Step 6: Write the Locus Equation
This equation represents the locus of point \( P(h, k) \):
\[ h(a_2 - a_1) + k(b_2 - b_1) = \frac{1}{2}(a_2^2 + b_2^2 - a_1^2 - b_1^2) \]

### Step 7: Compare with Given Equation
The given equation is:
\[ (a_1 - a_2)x + (b_1 + b_2)y + c = 0 \]
To match the form, we can rearrange the locus equation:
\[ (a_1 - a_2)x + (b_1 - b_2)y + c = 0 \]
From the comparison, we can identify \( c \):
\[ c = -\frac{1}{2}(a_2^2 + b_2^2 - a_1^2 - b_1^2) \]

### Final Answer
Thus, the value of \( c \) is:
\[ c = \frac{1}{2}(a_1^2 + b_1^2 - a_2^2 - b_2^2) \]

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