Key Concepts and Formulas

• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Locus: The set of all points that satisfy a given condition.
• Perpendicular Bisector: The locus of a point equidistant from two given points is the perpendicular bisector of the line segment joining the two points.

Step-by-Step Solution

Step 1: Define the Point and Distances Let $P(x, y)$ be a point on the locus. Let $A(a_1, b_1)$ and $B(a_2, b_2)$ be the two given points. Since $P$ is equidistant from $A$ and $B$, we have $PA = PB$.

Step 2: Apply the Distance Formula Using the distance formula, we have: $PA = \sqrt{(x - a_1)^2 + (y - b_1)^2}$ $PB = \sqrt{(x - a_2)^2 + (y - b_2)^2}$ Since $PA = PB$, 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 and Expand Squaring both sides to eliminate the square roots, we get: $(x - a_1)^2 + (y - b_1)^2 = (x - a_2)^2 + (y - b_2)^2$ Expanding the squares, we have: $x^2 - 2a_1x + a_1^2 + y^2 - 2b_1y + b_1^2 = x^2 - 2a_2x + a_2^2 + y^2 - 2b_2y + b_2^2$

Step 4: Simplify the Equation We can cancel out the $x^2$ and $y^2$ terms from both sides: $- 2a_1x + a_1^2 - 2b_1y + b_1^2 = - 2a_2x + a_2^2 - 2b_2y + b_2^2$ Rearranging the terms, we get: $2a_2x - 2a_1x + 2b_2y - 2b_1y + a_1^2 + b_1^2 - a_2^2 - b_2^2 = 0$ Factoring out the $x$ and $y$ terms, we have: $2(a_2 - a_1)x + 2(b_2 - b_1)y + (a_1^2 + b_1^2 - a_2^2 - b_2^2) = 0$

Step 5: Match the Given Equation Form We are given that the equation of the locus is $(a_1 - a_2)x + (b_1 - b_2)y + c = 0$. We need to rewrite our equation in this form. Multiplying our derived equation by $-\frac{1}{2}$, we get: $-(a_2 - a_1)x - (b_2 - b_1)y - \frac{1}{2}(a_1^2 + b_1^2 - a_2^2 - b_2^2) = 0$ $(a_1 - a_2)x + (b_1 - b_2)y + \frac{1}{2}(a_2^2 + b_2^2 - a_1^2 - b_1^2) = 0$

Step 6: Determine the Value of 'c' Comparing this with the given form $(a_1 - a_2)x + (b_1 - b_2)y + c = 0$, we can identify $c$ as: $c = \frac{1}{2}(a_2^2 + b_2^2 - a_1^2 - b_1^2)$

Common Mistakes & Tips

• Be careful with signs when expanding the squared terms and rearranging the equation. A small sign error can lead to an incorrect answer.
• Remember that the locus is the perpendicular bisector. This knowledge can help you check if your final equation makes sense geometrically.
• Double-check your algebraic manipulations to ensure accuracy.

Summary The equation of the locus of a point equidistant from two fixed points $(a_1, b_1)$ and $(a_2, b_2)$ is derived by setting the distances from the moving point $(x,y)$ to each fixed point equal. After careful algebraic expansion and rearrangement to match the specified linear equation form $(a_1 - a_2)x + (b_1 - b_2)y + c = 0$, the constant term $c$ is found to be $\frac{1}{2}(a_2^2 + b_2^2 - a_1^2 - b_1^2)$.

The final answer is \boxed{{1 \over 2}\left( {{a_2}^2 + {b_2}^2 - {a_1}^2 - {b_1}^2} \right)}, which corresponds to option (B).