How To Find The Center and Radius of a Circle

For an equation like (x - 3)^2 + (y - 4)^2 = 25, 'h' is found by reversing the sign of the number next to 'x' (so -3 becomes +3), and 'k' by reversing the sign of the number next to 'y' (so -4 becomes +4). Therefore, the center is (3, 4). The radius 'r' is the square root of the number on the right side of the equation (sq. root of 25 is 5).

A second example demonstrates with x + 2 squared + y - 5 squared = 49. Here, 'h' is -2 (from +2) and 'k' is +5 (from -5), making the center (-2, 5). The radius is the square root of 49, which is 7.

The tutorial then addresses more complex scenarios where the circle's equation is not in standard form. The key method introduced is 'completing the square'. This involves rearranging terms, grouping x-terms and y-terms, and moving constant terms to the right side of the equation. To complete the square, take half of the coefficient of the x-term (or y-term) and square it, adding this value to both sides of the equation.

After completing the square, the expressions can be factored into the (x - h)^2 and (y - k)^2 forms. For example, x^2 - 8x + 16 factors to (x - 4)^2. Once factored and simplified, the equation is in standard form, allowing for easy identification of the center (h, k) and the radius (r) by taking the square root of the constant on the right side.

The video begins by introducing the standard form of a circle's equation: (x - h)^2 + (y - k)^2 = r^2. In this equation, (h, k) represents the center of the circle, and 'r' is the radius.