A cone is a surface generated by a straight line which passes through a fixed point and intersects a fixed curve or touches a given curve.
The fixed point is called the vertex of the cone and the fixed curve is called a guiding curve of the cone. The straight line is called a generator.
Let (α, β, γ) be the given vertex and ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, z = 0 be the guiding curve.
The equations of any line passing through the point (α, β, γ) are
When this line meets the plane at z = 0 we get,
This point lies on the given curve ...