To find the equation of a circle, we use the standard formula:
(x – h)² + (y – k)² = r²
In this formula, (h, k) is the center of the circle, and r is the radius. Here, we know the center, which is (2, 4).
First, we substitute the center coordinates into the equation:
(x – 2)² + (y – 4)² = r²
Next, we need to find the radius (r). The radius can be calculated as the distance from the center of the circle to any point on the circle. In this case, we will use the point (7, 1) that lies on the circle.
To find the distance (r), we apply the distance formula:
r = √((x₂ – x₁)² + (y₂ – y₁)²)
Here, (x₁, y₁) is the center (2, 4) and (x₂, y₂) is the point (7, 1):
r = √((7 – 2)² + (1 – 4)²)
Now, calculating the differences:
r = √(5² + (-3)²) = √(25 + 9) = √34
Now we can substitute the value of r² into our circle equation:
(x – 2)² + (y – 4)² = 34
This is the equation of the circle that passes through the point (7, 1) with a center at (2, 4).