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
Step 1: Define the coordinates and the given condition. Let the coordinates of point P be (2, 5), point Q be (4, 7), and the variable point R be (x, y). The given condition is PR = 3QR. Step 2: Write the distance formulas for PR and QR. The distance formula between two points (x_a, y_a) and (x_b, y_b) is sqrt((x_b - x_a)^2 + (y_b - y_a)^2). PR = sqrt((x - 2)^2 + (y - 5)^2) QR = sqrt((x - 4)^2 + (y - 7)^2) Step 3: Substitute the distance formulas into the given condition and square both sides. PR = 3QR sqrt((x - 2)^2 + (y - 5)^2) = 3sqrt((x - 4)^2 + (y - 7)^2) Squaring both sides: (x - 2)^2 + (y - 5)^2 = 9[(x - 4)^2 + (y - 7)^2] Step 4: Expand and simplify the equation. Expand the squared terms: (x^2 - 4x + 4) + (y^2 - 10y + 25) = 9[(x^2 - 8x + 16) + (y^2 - 14y + 49)] x^2 + y^2 - 4x - 10y + 29 = 9[x^2 + y^2 - 8x - 14y + 65] Distribute the 9 on the right side: x^2 + y^2 - 4x - 10y + 29 = 9x^2 + 9y^2 - 72x - 126y + 585 Step 5: Rearrange the terms to form the equation of the locus. Move all terms to one side of the equation: 0 = 9x^2 - x^2 + 9y^2 - y^2 - 72x + 4x - 126y + 10y + 585 - 29 0 = 8x^2 + 8y^2 - 68x - 116y + 556 Divide the entire equation by 4 to simplify: 0 = (8x^2)/(4) + (8y^2)/(4) - (68x)/(4) - (116y)/(4) + (556)/(4) 0 = 2x^2 + 2y^2 - 17x - 29y + 139 The equation of the locus of R is 2x^2 + 2y^2 - 17x - 29y + 139 = 0. 3 done, 2 left today. You're making progress.