(b) \[ y^{2}-18 y=8 x-9^{2} \] vertex: focus : directrix"

Step 1 ：Given equation is \(y^2 - 18y = 8x - 81\)

Step 2 ：Rearrange the terms to get \(y^2 - 18y + 81 = 8x\)

Step 3 ：This can be rewritten as \((y - 9)^2 = 8x\)

Step 4 ：This is now in the standard form of a parabola that opens to the right. The vertex \((h, k)\) of the parabola is the point \((0, 9)\)

Step 5 ：The value of \(4a\) is 8, so \(a = 2\). The focus of the parabola is \(a\) units to the right of the vertex, so the focus is at the point \((2, 9)\)

Step 6 ：The directrix of the parabola is a vertical line \(a\) units to the left of the vertex, so the directrix is the line \(x = -2\)

Step 7 ：So, the vertex of the parabola is \(\boxed{(0, 9)}\), the focus is \(\boxed{(2, 9)}\), and the directrix is the line \(\boxed{x = -2}\)