+617 Find the area of the region enclosed by the graph of $x^2 + y^2 = 2x - 6y + 6 + 14x - 16y + 80$.
+1804 First, let's simplify the equation a bit. We get
\(x^2 - 16x + y^2 + 22y = 86\)
Completing the square for BOTH x and y, we get the equation
\(x^2 - 16x + 64 + y^2 + 22y + 121 = 86 + 64 + 121 \\ (x - 8)^2 + (y + 11)^2 = 271\)
This is a circle centered at (8, - 11) with r^2 = 217
Now, we can find the area. We simply have
\(Area = pi * r^2 = 271 pi ≈ 851.4\)
Thanks! :)
