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: Identify the coordinates of the endpoints of the diameter. Given the endpoints of the diameter are A(x_1, y_1) = (2,4) and B(x_2, y_2) = (-2,6). Step 2: Use the diameter form of the equation of a circle. The equation of a circle with endpoints of a diameter (x_1, y_1) and (x_2, y_2) is given by: (x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0 Step 3: Substitute the given coordinates into the equation. Substitute x_1=2, y_1=4, x_2=-2, and y_2=6: (x-2)(x-(-2)) + (y-4)(y-6) = 0 (x-2)(x+2) + (y-4)(y-6) = 0 Step 4: Expand and simplify the equation. Expand the first product (x-2)(x+2): x^2 - (2)^2 = x^2 - 4 Expand the second product (y-4)(y-6): y^2 - 6y - 4y + (-4)(-6) = y^2 - 10y + 24 Substitute these expanded forms back into the equation: (x^2 - 4) + (y^2 - 10y + 24) = 0 x^2 + y^2 - 10y - 4 + 24 = 0 x^2 + y^2 - 10y + 20 = 0 The equation of the circle is x^2 + y^2 - 10y + 20 = 0.