Question

which equation represents the circle described?
the radius is 2 units.
the center is the same as the center of a circle whose equation is (x^2 + y^2 - 8x - 6y + 24 = 0)
((x + 4)^2 + (y + 3)^2 = 2)
((x - 4)^2 + (y - 3)^2 = 2)
((x - 4)^2 + (y - 3)^2 = 2^2)
((x + 4)^2 + (y + 3)^2 = 2^2)

Explanation:

Step1: Find center of given circle

Rewrite \(x^2 + y^2 - 8x - 6y + 24 = 0\) by completing squares:
\((x^2 - 8x) + (y^2 - 6y) = -24\)
\((x-4)^2 - 16 + (y-3)^2 - 9 = -24\)
\((x-4)^2 + (y-3)^2 = 1\). Center is \((4,3)\).

Step2: Match center and radius

Standard circle equation: \((x-h)^2 + (y-k)^2 = r^2\) (center \((h,k)\), radius \(r\)). Need center \((4,3)\) and radius \(2\) (\(r^2=4\)).
Check options: Only \((x-4)^2 + (y-3)^2 = 2^2\) fits.

Answer:

\((x - 4)^2 + (y - 3)^2 = 2^2\) (the second option)