Final answer:
To determine the equation of the circle, we use the center coordinates and the distance from the center to any point on the circle. The equation is (x - 2)^2 + (y - 3)^2 = 58.
Step-by-step explanation:
To determine the equation of a circle, we need the coordinates of the center and the distance from the center to any point on the circle. In this case, the given center coordinates are (2,3) and the given point on the circle is (5,-4). The distance between the center and the point can be found using the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Plugging in the values, we get:
d = sqrt((5 - 2)^2 + (-4 - 3)^2)
d = sqrt(3^2 + (-7)^2)
d = sqrt(9 + 49)
d = sqrt(58)
So, the distance between the center and the point is sqrt(58).
Now, we can write the equation of the circle using the standard form:
(x - h)^2 + (y - k)^2 = r^2
Substituting the values, we get:
(x - 2)^2 + (y - 3)^2 = sqrt(58)^2
(x - 2)^2 + (y - 3)^2 = 58