To find the equation of the locus of a point \( P(x,y) \) such that the distance from \( P \) to point \( V(1,1) \) is equal to the distance from \( P \) to point \( W(3,5) \), we start by using the distance formula. The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] ### Step 1: Set up the equation 1. **Distance from \( P \) to \( V \)**: \[ PV = \sqrt{(x - 1)^2 + (y - 1)^2} \] 2. **Distance from \( P \) to \( W \)**: \[ PW = \sqrt{(x - 3)^2 + (y - 5)^2} \] ### Step 2: Set the distances equal Since we want \( PV = PW \), we can set the two distance expressions equal to each other: \[ \sqrt{(x - 1)^2 + (y - 1)^2} = \sqrt{(x - 3)^2 + (y - 5)^2} \] ### Step 3: Square both sides To eliminate the square roots, we square both sides: \[ (x - 1)^2 + (y - 1)^2 = (x - 3)^2 + (y - 5)^2 \] ### Step 4: Expand both sides Now we expand both sides: - Left side: \[ (x - 1)^2 + (y - 1)^2 = (x^2 - 2x + 1) + (y^2 - 2y + 1) = x^2 + y^2 - 2x - 2y + 2 \] - Right side: \[ (x - 3)^2 + (y - 5)^2 = (x^2 - 6x + 9) + (y^2 - 10y + 25) = x^2 + y^2 - 6x - 10y + 34 \] ### Step 5: Set the expanded forms equal Now we have: \[ x^2 + y^2 - 2x - 2y + 2 = x^2 + y^2 - 6x - 10y + 34 \] ### Step 6: Simplify the equation Subtract \( x^2 + y^2 \) from both sides: \[ -2x - 2y + 2 = -6x - 10y + 34 \] Now, rearranging gives: \[ -2x + 6x - 2y + 10y = 34 - 2 \] This simplifies to: \[ 4x + 8y = 32 \] ### Step 7: Divide through by 4 To simplify further, divide the entire equation by 4: \[ x + 2y = 8 \] ### Conclusion: Identify the correct option The equation of the locus is: \[ x + 2y = 8 \] Thus, the correct option is **D**. ### Explanation of Other Options - **Option A: \( 2x + 2y = 9 \)**: This does not satisfy the derived equation. If we rearranged it, we would find that it does not represent the same line as \( x + 2y = 8 \). - **Option B: \( 2x + 3y = 8 \)**: This is a different linear equation and does not match our derived equation. - **Option C: \( 2x + y = 9 \)**: Again, this is a different linear equation and does not represent the same relationship as \( x + 2y = 8 \). ### Revision Summary - The locus of points equidistant from two fixed points is a straight line. - Use the distance formula to set up the equation based on the distances to the two points. - Square both sides to eliminate square roots, then expand and simplify. - The final equation can be simplified to find the locus equation. - The correct answer is \( x + 2y = 8 \) (Option D).