find the necessary coordinates of 9x^2-4y^2-18x+8y+41=0
I don't know what you mean by necessary, but the hyperbola is
9x^2+18x - 4y^2+8y = -41
9(x^2+2x+1) - 4(y^2-2y+1) = -41 + 9*1 - 4*1
9(x+1)^2 - 4(y-1)^2 = -36
or,
4(y-1)^2 - 9(x+1)^2 = 36
(y-1)^2/9 - (x+1)^2/4 = 1
That should tell you what you need to know.