Step 1: Understanding the Question:
The objective of this problem is to determine the general equation of a circle given its center and one point lying on its circumference.
A circle is defined by its center \( (h, k) \) and its radius \( r \).
The radius is the distance from the center to any point on the boundary of the circle.
Once we have both the center and the radius, we can write the equation in central form and then expand it into the general form \( x^2 + y^2 + 2gx + 2fy + c = 0 \). Step 2: Key Formula or Approach:
1. Use the distance formula to find the radius \( r \): \( r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
2. Use the standard form of the circle equation: \( (x - h)^2 + (y - k)^2 = r^2 \).
3. Expand and rearrange the equation to the general form. Step 3: Detailed Explanation:
Calculating the radius squared (\( r^2 \)):
The center is given as \( (h, k) = (2, -1) \) and the circle passes through \( (3, 6) \).
By applying the distance formula between these two points:
\[ r^2 = (3 - 2)^2 + (6 - (-1))^2 \]
\[ r^2 = (1)^2 + (7)^2 \]
\[ r^2 = 1 + 49 = 50 \]
Writing the equation in standard form:
Substituting the center \( (2, -1) \) and \( r^2 = 50 \) into the equation \( (x - h)^2 + (y - k)^2 = r^2 \):
\[ (x - 2)^2 + (y - (-1))^2 = 50 \]
\[ (x - 2)^2 + (y + 1)^2 = 50 \]