Easy!
1. Imagine a Cartesian coordinate system with the origin located where the point of the cone would be. Have the z-axis parallel with the centre-line of the side pipe.
2. The centre-line of the side pipe is located at x0,y0, and the side pipe has radius r0. The side pipe has an equation something like: (x-x0)^2 + (y-y0)^2 - r0^2 = 0.
3. If the cone widens at a rate k, then the equation for the surface of the cone is x^2 + z^2 - (ky)^2 = 0.
4. Solve these two equations to give you another equation for the line where the two parts intersect. Should be something like: x0^2 - 2x*x0 - z^2 +(k^2 -1)y^2 - 2y*y0 + y^2 - r0^2 = 0.
5. Create a coordinate system based on the flat sheet, with coordinates r and a. These are related to the Cartesian coordinate system by y = r/(1+k^2)^0.5, x=r cos(a) and z=r sin(a). r is the radial coordinate on the flat sheet, and a is an angular coordinate.
6. Now express the intersect in terms of r and a. Something like: x0^2 - 2x0r cos(a) - r^2 sin^2(a) + (k^2 - 1)r^2 / (1+k^2) - 2y0r/(1+k^2)^0.5 + r^2 /(1+k^2) - r0 = 0 will be the equation of the line you need to cut on your flat sheet.
7. Remember the cone is rotationally symmetric, so you can shift coordinate a by any amount a0, to avoid cutting on the join. Also, the above equations should have 2 solutions, as I haven't specified a finite length side pipe (ie. don't cut 2 holes).
On reflection, the methods suggested by others might be better...
Post edited at 10:31