Given the endpoints of the diameter are A(x1, y1) = (2,4) and B(x2, y2) = (-2,6).
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 $\boxed{x^2 + y^2 - 10y + 20 = 0}$.
