Angle between two intersecting surfaces

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Problem 3559 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.5 "Applications in Geometry"

At what angle does the cylinder x2+y2=a2x^{2}+y^{2}=a^{2} intersect the surface bz=xybz=xy at a common point M0(x0,y0,z0)M_{0}(x_{0}, y_{0}, z_{0})?

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Problem 3.6.10 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.6.2 "Problems"

Prove that the surfaces z=xy−x2+8x−5,z=ex+2y+4z = x y-x^{2}+8 x-5, \quad z = e^{x+2 y+4} are tangent to each other at the point (2;−3;1)(2 ;-3 ; 1), and find the equation of the common tangent plane.

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Problem 3.6.27 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.6.2 "Problems"

Determine the angle at which the surfaces intersect:

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(1)

z2=xy,x2+y2+z2=1z^{2} = x y, x^{2}+y^{2}+z^{2} = 1;

(2)

xy=az,x2+y2=b2,b>0x y = a z, x^{2}+y^{2} = b^{2}, b > 0;

(3)

xy=az,x2+z2+y2+z2=b,b>0x y = a z, \sqrt{x^{2}+z^{2}}+\sqrt{y^{2}+z^{2}} = b, b > 0.

Problem 3.6.28 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.6.2 "Problems"

Prove the orthogonality of the surfaces

xyz=a3 and 2z2=x2+y2+f(x2−y2) x y z = a^{3} \quad \text{ and } \quad 2 z^{2} = x^{2}+y^{2}+f\left(x^{2}-y^{2}\right)

where f(u)f(u) is a differentiable function.

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References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.
[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Функции многих переменных. 2-е изд., ФИЗМАТЛИТ, 2003.