Write the equation for a circle centered at c (2; 1) through d (5; 5).

The equation of a circle of radius R centered at the point C (a; b) has the form:
(x – a) ² + (y – b) ² = R².
1. Radius – the distance from the center of the circle to any point on the circle. Thus, the radius will be equal to the distance from point c (2; 1) to point d (5; 5).
The distance between points A (x₁; y₁) and B (x₂; y₂) is calculated by the formula:
AB = √ ((x₁ – x₂) ² + (y₁ – y₂) ²).
Thus, the distance between points c (2; 1) and d (5; 5) will be equal to:
cd = R = √ ((2 – 5) ² + (1 – 5) ²) = √ ((- 3) ² + (- 4) ²) = √ (9 + 16) = √25 = 5.
1. Substitute the known values ​​into the equation of a circle of radius R = 5 centered at point c (2; 1):
(x – 2) ² + (y – 1) ² = 5²;
(x – 2) ² + (y – 1) ² = 25.
Answer: (x – 2) ² + (y – 1) ² = 25.