Identities for the normal vector and tangent plane of a parametrized surface

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Problem 2274 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 5.8 "Space Curves: The Osculating Plane, the Principal Normal, and the Binormal"

Prove that the osculating plane of the curve

x2=φ′(t),y2=t2φ′(t),z=φ(t), x^{2} = \varphi '(t), \qquad y^{2} = t^{2}\varphi '(t), \qquad z = \varphi (t),

lying on the conoidal surface z=φ(yx)z=\varphi \left(\dfrac {y}{x}\right), coincides with the tangent plane to this surface.

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Problem 1.10.30 from Rudin W. "Principles of Mathematical Analysis" [Rud76]More problems in Chapter 1.10 "Integration of Differential Forms"

If N\mathbf{N} is the vector given by (135), prove that

det⁡[α1β1α2β3−α3β2α2β2α3β1−α1β3α3β3α1β2−α2β1]=∣N∣2 \operatorname {det}\left[\begin{smallmatrix} \alpha _{1} & \beta _{1} & \alpha _{2} \beta _{3}-\alpha _{3} \beta _{2} \\ \alpha _{2} & \beta _{2} & \alpha _{3} \beta _{1}-\alpha _{1} \beta _{3} \\ \alpha _{3} & \beta _{3} & \alpha _{1} \beta _{2}-\alpha _{2} \beta _{1} \end{smallmatrix}\right] = \left|\mathbf{N}\right|^{2}

Also, verify Eq. (137).

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References

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.
[Rud76] Rudin W. Principles of Mathematical Analysis. 3rd ed., McGraw-Hill, 1976.