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: Write the standard equation of a circle. The standard equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2. Given the center D is (2, -1), so h = 2 and k = -1. Substituting these values, the equation becomes: (x - 2)^2 + (y - (-1))^2 = r^2 (x - 2)^2 + (y + 1)^2 = r^2 Step 2: Calculate the radius squared (r^2). The circle passes through point P (-6, 5). The distance between the center D and point P is the radius r. We can use the distance formula d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) or directly calculate r^2 using the points. r^2 = (-6 - 2)^2 + (5 - (-1))^2 r^2 = (-8)^2 + (5 + 1)^2 r^2 = (-8)^2 + (6)^2 r^2 = 64 + 36 r^2 = 100 Step 3: Substitute r^2 into the standard equation. (x - 2)^2 + (y + 1)^2 = 100 Step 4: Expand the equation to the general form x^2 + y^2 + ax + by + c = 0. Expand (x - 2)^2: (x - 2)^2 = x^2 - 2(x)(2) + 2^2 = x^2 - 4x + 4 Expand (y + 1)^2: (y + 1)^2 = y^2 + 2(y)(1) + 1^2 = y^2 + 2y + 1 Substitute these expanded forms back into the equation: (x^2 - 4x + 4) + (y^2 + 2y + 1) = 100 Combine like terms and move the constant to the left side: x^2 + y^2 - 4x + 2y + 4 + 1 - 100 = 0 x^2 + y^2 - 4x + 2y - 95 = 0 The equation of the circle is x^2 + y^2 - 4x + 2y - 95 = 0. 3 done, 2 left today. You're making progress.