(x - 2)² + (y + 5)² = 49. The equation of the circle in standard form from the equation x² - 4x + y² + 10y -20 = 0 is (x - 2)² + (y + 5)² = 49, where the center c(2, -5) and the radius r = 7.
The key to solve this problem given the general form equation x² + y² + Ax + By + c= 0, it will be convert into the standard form equation (x - h)² + (y - k)² = r² to determinate the center, radius, and to graph the circle.
Given the general form equation x² - 4x + y² + 10y -20 = 0, to convert to the standard form equation (x - h)² + (y - k)² = r², we have to follow a steps:
1. Group the terms related to x and y. Then, move any numerical constant to the right side of the equation.
In this case, the general form equation it's already group by related terms of x and y, we only have to move the numerical constant.
x² - 4x + y² + 10y = 20
2. We need to create a empty space to form perfect square trinomials.
x² - 4x + ( ) + y² + 10y + ( ) = 20 + ( ) + ( )
3. To find the missing values, taking the middle terms of x and y in the equation (b/2)², this is called completing the square.
For x, b = -4 -------> (b/2)² = (-4/2)² = (-2)² = 4
For y, b = 10 -------> (b/2)² = (10/2)² = (5)² = 25
Completing the square:
x² - 4x + (4) + y² + 10y + (25) = 20 + (4) + (25)
x² - 4x + 4 + y² + 10y + 25 = 20 + 4 + 25
x² - 4x + 4 + y² + 10y + 25 = 49
4. Rewriting in factored form.
(x - 2)² + (y + 5)² = 49
To find the center:
(x - h)² + (y - k)² = r² Where the center c is (h, k), and r² is the radius.
Solving for (x - 2)² + (y + 5)² = 49
The center is c(2, -5)
The radius is r² = 49, so r = √49 = 7
With this values you could easily graph the circle.
