Find the area of the region enclosed by the graph of x^2 + y^2 = 2x - 6y + 6 + 4x - 8y.
In order to find the area of this region, we have to first conver this equation into the standard circle equation. Moving all terms (except the constant term) to the left handside, and simplifying, we get:
\(x^2 - 6x + y^2 + 14y = 6\)
Then, we can complete the square by using the formula (b/2)^2, and add the coresponding value to both sides. Therefore, we obtain:
\((x-3)^2 + (y + 7)^2 = 64\)
Therefore, this is a circle with center (3, -7), and a radius of 8. Therefore, the area of this circle is \(64 \pi\)