Determine the equation of the circle with center left bracket, 5, comma, minus, 8, right bracket(5,−8) containing the point left bracket, 14, comma, minus, 17, right bracket(14,−17).
The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
In this case, the center is (5, -8) and the point (14, -17) lies on the circle. To find the radius, we can use the distance formula between the center and the point on the circle:
r = sqrt((14 - 5)^2 + (-17 - (-8))^2)
r = sqrt(9^2 + 9^2)
r = sqrt(162)
r = 9√2
Now we can plug in the values of the center and the radius into the equation of the circle:
(x - 5)^2 + (y + 8)^2 = (9√2)^2
(x - 5)^2 + (y + 8)^2 = 162
Therefore, the equation of the circle is:
(x - 5)^2 + (y + 8)^2 = 162