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: Rearrange the equation to group x-terms and y-terms. The given equation of the circle is x^2 - 6x + y^2 - 10y + 30 = 0. Group the x-terms and y-terms, and move the constant term to the right side of the equation: (x^2 - 6x) + (y^2 - 10y) = -30 Step 2: Complete the square for the x-terms. To complete the square for x^2 - 6x, take half of the coefficient of x (which is -6), square it, and add it to both sides of the equation. Half of -6 is -3. (-3)^2 = 9. (x^2 - 6x + 9) + (y^2 - 10y) = -30 + 9 Step 3: Complete the square for the y-terms. To complete the square for y^2 - 10y, take half of the coefficient of y (which is -10), square it, and add it to both sides of the equation. Half of -10 is -5. (-5)^2 = 25. (x^2 - 6x + 9) + (y^2 - 10y + 25) = -30 + 9 + 25 Step 4: Rewrite the equation in standard form. The expressions in parentheses are now perfect squares: (x - 3)^2 + (y - 5)^2 = 4 Step 5: Identify the center and radius. The standard form of the equation of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. Comparing our equation (x - 3)^2 + (y - 5)^2 = 4 with the standard form: The center (h, k) is (3, 5). The radius squared r^2 = 4, so the radius r = sqrt(4) = 2. The center of the circle is (3, 5). The radius of the circle is 2. 3 done, 2 left today. You're making progress.