Coordinate Geometry - Finding the Centre and Radius of a Circle

Given that the equation of the circle is \(x^{2} + y^{2} + 12x - 16y + k = 0\) where \(k\) is a constant.
Determine the value of \(k\) for each of the following cases:

1. The area of the circle is \(81\pi\) square units. Also, state the centre of the circle.
2. The line \(y = x\) is a tangent to the circle.