- Identify that the right angle at \(A\) means \(BC\) is the diameter of the circumcircle.
- Calculate the center \((h, k)\) as the midpoint of \(BC\), which is \((2, -4)\).
- Determine the squared radius \(r^2\) as the distance from the center to vertex \(A\), resulting in \(50\).
- Expand the standard form equation \((x - 2)^2 + (y + 4)^2 = 50\) to the general form \(\boxed{x^2 + y^2 - 4x + 8y - 30 = 0}\).
Explanation
- Analyze the geometry of the triangle and the circle.
The points \(A(1, 3)\), \(B(7, 1)\), and \(C(-3, -9)\) form a triangle. Since \(\triangle ABC\) is right-angled at \(A\), the hypotenuse \(BC\) serves as the diameter of the circumcircle. This is a key property of right-angled triangles: the circumcenter is the midpoint of the hypotenuse, and the distance from the center to any vertex is the radius.
- Calculate the center of the circle.
To find the center \((h, k)\) of the circle, we calculate the midpoint of the diameter \(BC\):
\[h = \frac{x_B + x_C}{2} = \frac{7 + (-3)}{2} = \frac{4}{2} = 2\]
\[k = \frac{y_B + y_C}{2} = \frac{1 + (-9)}{2} = \frac{-8}{2} = -4\]
The center of the circle is \((2, -4)\).
- Calculate the squared radius of the circle.
The squared radius \(r^2\) is the square of the distance between the center \((2, -4)\) and point \(A(1, 3)\):
\[r^2 = (x_A - h)^2 + (y_A - k)^2\]
\[r^2 = (1 - 2)^2 + (3 - (-4))^2\]
\[r^2 = (-1)^2 + (7)^2 = 1 + 49 = 50\]
Thus, the squared radius is \(50\).
- Derive the general equation of the circle.
The standard form of the circle equation is \((x - h)^2 + (y - k)^2 = r^2\). Substituting our values:
\[(x - 2)^2 + (y + 4)^2 = 50\]
Expanding the terms:
\[(x^2 - 4x + 4) + (y^2 + 8y + 16) = 50\]
\[x^2 + y^2 - 4x + 8y + 20 = 50\]
Subtracting 50 from both sides gives the general form:
\[x^2 + y^2 - 4x + 8y - 30 = 0\]
The equation of the circle is \(\boxed{x^2 + y^2 - 4x + 8y - 30 = 0}\).
Examples
Engineers and surveyors often use the properties of circles to determine the layout of circular structures such as roundabouts or circular containment areas. Knowing that a right-angled structure (like two perpendicular fence lines) can be inscribed in a circle, they can pinpoint the exact center and size of the required circular area by identifying the two endpoints of the diameter. For example, if two construction points \(B\) and \(C\) are known, the circle's equation helps them calculate the precise location for drilling or construction equipment at any point along the arc.