This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — Step 1: Write the distance formulas for |PR| and |QR|. Let R(x, y) be the variable point. The fixed points are P(2, 5) and Q(4, 7). The distance formula between two points (x_1, y_1) and (x_2, y_2) is sqrt((x_2-x_1)^2 + (y_2-y_1)^2). For |PR|: |PR| = sqrt((x-2)^2 + (y-5)^2) For |QR|: |QR| = sqrt((x-4)^2 + (y-7)^2) Step 2: Substitute these expressions into the given condition 5|PR| = 3|QR|. 5sqrt((x-2)^2 + (y-5)^2) = 3sqrt((x-4)^2 + (y-7)^2) Step 3: Square both sides of the equation to eliminate the square roots. (5sqrt((x-2)^2 + (y-5)^2))^2 = (3sqrt((x-4)^2 + (y-7)^2))^2 25((x-2)^2 + (y-5)^2) = 9((x-4)^2 + (y-7)^2) Step 4: Expand the squared terms inside the parentheses. (x-2)^2 = x^2 - 4x + 4 (y-5)^2 = y^2 - 10y + 25 (x-4)^2 = x^2 - 8x + 16 (y-7)^2 = y^2 - 14y + 49 Substitute these expansions back into the equation: 25(x^2 - 4x + 4 + y^2 - 10y + 25) = 9(x^2 - 8x + 16 + y^2 - 14y + 49) 25(x^2 - 4x + y^2 - 10y + 29) = 9(x^2 - 8x + y^2 - 14y + 65) Step 5: Distribute the constants and simplify. 25x^2 - 100x + 25y^2 - 250y + 725 = 9x^2 - 72x + 9y^2 - 126y + 585 Step 6: Move all terms to one side to form the equation of the locus. Subtract 9x^2, -72x, 9y^2, -126y, and 585 from both sides: (25x^2 - 9x^2) + (-100x + 72x) + (25y^2 - 9y^2) + (-250y + 126y) + (725 - 585) = 0 16x^2 - 28x + 16y^2 - 124y + 140 = 0 Step 7: Divide the entire equation by the common factor, 4, to simplify. (16x^2)/(4) - (28x)/(4) + (16y^2)/(4) - (124y)/(4) + (140)/(4) = 0 4x^2 - 7x + 4y^2 - 31y + 35 = 0 The equation of the locus of R is: 4x^2 + 4y^2 - 7x - 31y + 35 = 0 Drop the next question.