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 standard equation of a circle is (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center and r is the radius. The given equation for number 8 is: (2x-2)^2 + 4(-y-3)^2 = 12 Step 1: Rewrite the terms to match the standard form. For the first term, factor out 2: (2x-2)^2 = (2(x-1))^2 = 2^2(x-1)^2 = 4(x-1)^2 For the second term, factor out -1 from (-y-3): (-y-3)^2 = (-(y+3))^2 = (-1)^2(y+3)^2 = (y+3)^2 Substitute these back into the equation: 4(x-1)^2 + 4(y+3)^2 = 12 Step 2: Divide the entire equation by 4 to get the coefficients of the squared terms to be 1. (4(x-1)^2)/(4) + (4(y+3)^2)/(4) = (12)/(4) (x-1)^2 + (y+3)^2 = 3 Step 3: Identify the center and radius from the standard form. Comparing (x-1)^2 + (y+3)^2 = 3 with (x-h)^2 + (y-k)^2 = r^2: The center (h,k) is (1, -3). The radius squared r^2 is 3, so the radius r is sqrt(3). The radius is sqrt(3) The coordinates of the center are (1, -3) 3 done, 2 left today. You're making progress.