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
The center of the circle is (1, 3/4) and the radius is sqrt(217) / 4. Here are the steps to determine the center and radius: 1. Expand and simplify the given equation: The equation is 1/2x(4x-8) + 1/4y(8y-12) = 24. Expand the terms: (1/2x)(4x) - (1/2x)(8) + (1/4y)(8y) - (1/4y)(12) = 24 2x^2 - 4x + 2y^2 - 3y = 24 2. Divide by the coefficient of x² and y² to make them 1: Divide the entire equation by 2: x^2 - 2x + y^2 - (3/2)y = 12 3. Rearrange terms and complete the square for x and y: Group the x-terms and y-terms: (x^2 - 2x) + (y^2 - (3/2)y) = 12 To complete the square for x^2 - 2x, take half of the coefficient of x (-2), which is -1, and square it (-1)^2 = 1. To complete the square for y^2 - (3/2)y, take half of the coefficient of y (-3/2), which is -3/4, and square it (-3/4)^2 = 9/16. Add these values to both sides of the equation: (x^2 - 2x + 1) + (y^2 - (3/2)y + 9/16) = 12 + 1 + 9/16 4. Rewrite in standard form of a circle (x-h)² + (y-k)² = r²: (x - 1)^2 + (y - 3/4)^2 = 13 + 9/16 Calculate the right side: 13 + 9/16 = (13 * 16)/16 + 9/16 = 208/16 + 9/16 = 217/16 So the equation in standard form is: (x - 1)^2 + (y - 3/4)^2 = 217/16 5. Identify the center and radius: Comparing this to (x-h)^2 + (y-k)^2 = r^2: The center (h, k) is (1, 3/4). The radius squared r^2 is 217/16. The radius r is sqrt(217/16) = sqrt(217) / sqrt(16) = sqrt(217) / 4.