Loci traced by the tangent, normal and binormal lines

The coordinates of the points of a certain curve satisfy the relation

Prove that the tangents to this curve are tangent to the sphere

Prove that the tangents to the curve

meet the plane $xOy$ in the circle $x^{2}+y^{2}=4a^{2}$.

Prove that one of the bisectors of the angles between the tangent and the binormal to the curve

has a constant direction.

Along the principal normals of the helix

segments of given length $l$ are laid off. Find the locus of their endpoints.

Find the surface on which all the principal normals of the helix of the preceding problem lie.

The same question (find the surface on which the lines lie) as in the preceding problem, for the binormals.

Prove that one of the bisectors of the angles between the tangent and the binormal to the curve $x = 3 t, y = 3 t^{2}, z = 2 t^{3}$ has a constant direction.

Along the principal normals (but in the opposite direction) of the helix $x = a \cos t, y = a \sin t, z = b t$, segments of length $l$ are laid off. Find the curve described by their endpoints.

Along the binormals of the helix $x = a \cos t, y = a \sin t, z = e^{t}$, segments of one and the same length are laid off. Find the equation of the curve formed by the endpoints of these segments.