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.

HumenBeing Jun 21, 2023

#1**+1 **

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)

= sqrt(25)

= 5

Therefore, the correct distance between the center of the circle and the point (1, 7) is indeed 5. myccpay

newsdop Jun 21, 2023