What is the distance between the center of the circle with equation x^2+y^2=-4x+6y-12 and the point (1,7)? I was pretty sure 10 is correct, but apparently, it is wrong. BTW I already finished the test, so it's not cheating.
EDIT: never mind I figured it out. For anyone who is wondering, the answer is 5.
The equation of the circle can be rewritten in the standard form by completing the square for both the x and y terms:
(x^2 + 4x) + (y^2 - 6y) = -12
To complete the square for the x terms, we need to add (4/2)^2 = 4 to both sides, and for the y terms, we need to add (-6/2)^2 = 9 to both sides:
(x^2 + 4x + 4) + (y^2 - 6y + 9) = -12 + 4 + 9
Simplifying, we have:
(x + 2)^2 + (y - 3)^2 = 1
Comparing this to the standard form of a circle equation, we find that the center of the circle is (-2, 3), and the radius is sqrt(1) = 1.
The distance between the center of the circle (-2, 3) and the point (1, 7) can be found using the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((1 - (-2))^2 + (7 - 3)^2)
= sqrt(3^2 + 4^2)
= sqrt(9 + 16)
Therefore, the correct distance between the center of the circle and the point (1, 7) is indeed 5. myccpay