- Identify that the hypotenuse \(BC\) of the right-angled triangle \(ABC\) is the diameter of its circumcircle.
- Determine the center \((h,k)\) of the circle as the midpoint of the diameter \(BC\), which is \((2, -4)\).
- Calculate the squared radius \(r^2\) using the distance formula from the center to one of the vertices, giving \(r^2 = 50\).
- Write the equation in the standard form \((x-2)^2 + (y+4)^2 = 50\) and expand it to the required general form \(\boxed{x^2 + y^2 - 4x + 8y - 30 = 0}\).
Explanation
- Identify the diameter of the circumcircle.
In a right-angled triangle, the circumcenter (the center of the circle passing through all three vertices) is the midpoint of the hypotenuse. We are given the vertices \(A(1,3)\), \(B(7,1)\), and \(C(-3,-9)\). Since the triangle is right-angled at \(A\), the side \(BC\) is the hypotenuse and serves as the diameter of the circumcircle.
- Find the center of the circle.
The center \((h,k)\) of the circle is the midpoint of the segment \(BC\). Using the midpoint formula \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\):
\(h = \frac{7 + (-3)}{2} = \frac{4}{2} = 2\)
\(k = \frac{1 + (-9)}{2} = \frac{-8}{2} = -4\)
The center of the circle is \((2, -4)\).
- Calculate the radius squared of the circle.
The radius squared \(r^2\) is the squared distance from the center \((2, -4)\) to any of the points on the circle, such as \(B(7, 1)\).
\(r^2 = (7 - 2)^2 + (1 - (-4))^2\)
\(r^2 = (5)^2 + (5)^2\)
\(r^2 = 25 + 25 = 50\)
Thus, the radius squared is 50.
- Determine the final equation of the circle.
The equation of a circle with center \((h,k)\) and radius squared \(r^2\) is given by the standard form:
\((x - h)^2 + (y - k)^2 = r^2\)
Substituting \(h = 2\), \(k = -4\), and \(r^2 = 50\):
\((x - 2)^2 + (y + 4)^2 = 50\)
Expanding the squares:
\((x^2 - 4x + 4) + (y^2 + 8y + 16) = 50\)
Combine like terms and set to zero to match the form \(x^2 + y^2 + ax + by + c = 0\):
\(x^2 + y^2 - 4x + 8y + 20 = 50\)
\(x^2 + y^2 - 4x + 8y - 30 = 0\)
The final equation of the circle is \(x^2 + y^2 - 4x + 8y - 30 = 0\).
Examples
Understanding how to find the equation of a circumcircle is essential in fields like computer-aided design (CAD) and robotics. For instance, when a robot arm needs to pass through three specific points in its workspace, engineers calculate the path using the circumcircle of those points to ensure smooth, circular motion. By knowing the coordinates of the three points, they can determine the required arc and program the path, ensuring the robot arm follows an optimal, controlled trajectory while maintaining accuracy.