To find the equation of a circle, we can use the standard form of a circle’s equation, which is given by:
(x – h)² + (y – k)² = r²
In this equation, (h, k) is the center of the circle and r is the radius. In our case, the center is given as (1, 4), so h = 1 and k = 4.
Next, we need to determine the radius ‘r’. The radius is the distance from the center of the circle to any point on the circle. We know that there is a point on the circle at (4, 8). We can find the radius using the distance formula:
r = √((x₂ – x₁)² + (y₂ – y₁)²)
Here, (x₁, y₁) is (1, 4) and (x₂, y₂) is (4, 8). Plugging in the values, we get:
r = √((4 – 1)² + (8 – 4)²)
r = √(3² + 4²)
r = √(9 + 16)
r = √25
r = 5
Now that we have the radius, we can substitute the values into the circle’s equation:
(x – 1)² + (y – 4)² = 5²
So, the equation of the circle becomes:
(x – 1)² + (y – 4)² = 25
This is the equation of the circle with a center at (1, 4) that passes through the point (4, 8).