Tangents and normals

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Problem: Find the equation of the tangent plane and the normal at a given point on the surface: \[ 1+\sqrt{x^{2}+y^{2}+z^{2}}=x+y+z, \quad M(2 ; 3 ; 6) . \]

3.4.1 Tangents and normals

2.54 $

Problem: Find the equation of the tangent plane to the surface \( x^{3}-2 x y+2 z^{2}=8 \) at the point \( A(2 ; 2 ; 2) \).

3.4.2 Tangents and normals

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Problem: Find the points of intersection of the straight line \( z=2 x=2 y \) and the surface \( x y=y+z \), as well as the angles between the straight line and the surface.

3.4.3 Tangents and normals

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Problem: Write the canonical equation of the straight line that is the intersection of planes: \( \Gamma x+\mathrm{H} y+3 z=\Gamma \) and \( x-y-\mathrm{Hz}=1 \), where \( \Gamma=\mathrm{H}=2 \).

3.4.4 Tangents and normals

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Problem: Find the angle between the straight line \( (A B) \) and the plane \( (C D E) \), where \( A(1 ; \Gamma ; 2), B(-1 ; H ; 1) \), \( C(\Gamma ; 0 ; 4), D(2 ; \mathrm{H} ; 1), E(\mathrm{H} ; \Gamma ; 3) \), where \( \Gamma=\mathrm{H}=2 \). First write the equations of the straight line and the plane.

3.4.5 Tangents and normals

1.78 $