To find the equation of a circle, we use the standard form:
(x – h)2 + (y – k)2 = r2
In this equation, (h, k) represents the center of the circle, and r is the radius. Here, the center of the circle is given as (1, 3).
Next, we need to find the radius of the circle. The radius can be calculated as the distance between the center of the circle (1, 3) and a point on the circle (7, 5). We use the distance formula:
d = √[(x2 – x1)2 + (y2 – y1)2]
Plugging in our values:
d = √[(7 – 1)2 + (5 – 3)2] = √[62 + 22] = √[36 + 4] = √40 = 2√10
Now that we have the radius (r = 2√10), we can plug the center coordinates and radius into the circle equation:
(x – 1)2 + (y – 3)2 = (2√10)2
This simplifies to:
(x – 1)2 + (y – 3)2 = 40
Therefore, the equation of the circle is:
(x – 1)2 + (y – 3)2 = 40