Before we begin solving for the center of this conic section, we should identify which conic section that we have. Since both x and y have square terms, we know that it is not a parabola. Since the coefficients on x2 and y2 are different, we know that it is not a circle. Since the sign on the x2 and y2 are opposite, we know it is not an ellipse. The conic section that we have is the hyperbola.
With this in mind, we need to rewrite the equation in standard form...
9x2 - 4y2 - 18x - 24y - 63 = 0
First, we are going to get the constant on the right by addind 63 to each side.
9x2 - 4y2 - 18x - 24y = 63
Next, we are going to group the x's and y's.
(9x2 - 18x) - (4y2 + 24y) = 63
Now, we factor out like coefficients in each group.
9(x2 - 2x) - 4(y2 + 6y) = 63
Next, we add the term in each parenthesis that completes the square (notice that it has to be multiplied by the coefficient when we add it to the right).
9(x2 - 2x + 1) -4(y2 + 6y + 9) = 63 + 9 - 36
Next, we factor the terms on the left into complete squares and simplify the right side.
9(x - 1)2 - 4(y + 3)2 = 36
Finally, we divide everything by the coefficients of the two terms on the left (9 and 4).
(x - 1)2/4 - (y + 3)2/9 = 1
The center of the conic is (1, -3), since 1 is subtracted from x and -3 is subtracted from y.