Curves crossing a ruled surface's generators at a constant angle

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Problem 2252 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.7 "Space Curves: The Tangent Line and the Normal Plane"

Prove that the curve

x = e^{t}\cos t, \qquad y = e^{t}\sin t, \qquad z = e^{t}

intersects all the generators of the cone $x^{2}+y^{2}=z^{2}$ at one and the same angle.

Problem 2253 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.7 "Space Curves: The Tangent Line and the Normal Plane"

For what $a$ does the curve

x = e^{at}\cos t, \qquad y = e^{at}\sin t, \qquad z = e^{at}

intersect all the generators of the cone $x^{2}+y^{2}=z^{2}$ at an angle $\dfrac {\pi }{4}$ ?

Problem 2254 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.7 "Space Curves: The Tangent Line and the Normal Plane"

Prove that the curves of intersection of the cylinders $y^{2}+z^{2}=m^{2}$ with the surface $xy=az$ intersect at a right angle all the generators of this surface belonging to one family.

Problem 2255 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.7 "Space Curves: The Tangent Line and the Normal Plane"

The curve $x=a\tg t$ , $y=b\cos t$ , $z=b\sin t$ lies on the surface of a paraboloid. Prove that it intersects all the generators of one family at a right angle.

Problem 2261 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.7 "Space Curves: The Tangent Line and the Normal Plane"

Prove that the tangent to the curve

x^{2} = 3y, \qquad 2xy = 9z

makes a constant angle with some fixed direction.

References

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.