91Ó°ÊÓ

Completing the square is a method used to convert a quadratic equation into a perfect square trinomial, making it easier to solve or to put into the standard form of a circle. In our exercise, we start with the equation: \[ x^2 + y^2 - 8x - 84 = 0 \]To complete the square, we first group the x-terms and y-terms, and then move the constant term to the right side of the equation: \[ x^2 - 8x + y^2 = 84 \]Next, focus on the x-terms: \[ x^2 - 8x \]Take half of the coefficient of x, which is -4, square it to get 16, and add it to both sides of the equation: \[ x^2 - 8x + 16 = 84 + 16 \]This forms a perfect square trinomial on the left and results in: \[ (x - 4)^2 + y^2 = 100 \]Now, we've successfully converted into a form that reveals important properties of the circle.

The center of a circle is a point that is equidistant from all points on the circle. In the standard form of a circle equation, \[ (x - h)^2 + (y - k)^2 = r^2 \]The center of the circle is given by the point (h, k). After completing the square in our exercise, we transformed our equation to: \[ (x - 4)^2 + y^2 = 100 \]From this, we can see that h = 4 and k = 0. Therefore, the center of our circle is (4, 0). This point is where we start when graphing our circle and it is essential for understanding its geometry.

The radius of a circle is the distance from the center to any point on the circle. It’s a key characteristic as it helps define the circle's size. In the standard form of a circle's equation \[ (x - h)^2 + (y - k)^2 = r^2 \]The radius is determined by taking the square root of the right side of the equation. From our example, we derived: \[ (x - 4)^2 + y^2 = 100 \]Here, r^2 equals 100, so the radius r is: \[ r = \sqrt{100} = 10 \]This means every point on the circle is 10 units away from the center (4, 0). This distance is crucial when drawing or understanding the circle.

The standard form of a circle equation helps easily identify the circle's center and radius. The equation is generally given as: \[ (x - h)^2 + (y - k)^2 = r^2 \]Where (h, k) represents the center and r represents the radius. In our exercise, after completing the square, we converted: \[ x^2 + y^2 - 8x - 84 = 0 \]into the standard form: \[ (x - 4)^2 + y^2 = 100 \]This form makes it straightforward to see that the center of the circle is (4, 0) and the radius is 10. Knowing this form is fundamental to solving any circle-related problems, as it directly provides the necessary geometric properties.