Surfaces of the 2-nd order

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Problem: Determine the type of the surface of the 2nd order, bringing the equation to the canonical form: \[ x^{2}+5 y^{2}+z^{2}+2 x y+6 x z+2 y z-6=0 \text {. } \]

3.5.1 Surfaces of the 2-nd order

2.5 $

Problem: Find out which surface is defined by the following equation and schematically depict it: \[ x^{2}+z^{2}-4 x-4 z+4=0 . \]

3.5.2 Surfaces of the 2-nd order

0.75 $

Problem: Find out which surface is defined by the following equation and schematically depict it: \[ x^{2}+2 y^{2}+z^{2}-2 x y-2 y z+4=0 \text {. } \]

3.5.3 Surfaces of the 2-nd order

0.75 $

Problem: Let the given surface be \( \sigma: 9(x-1)^{2}+4 y^{2}-36 z^{2}=36 \) and the points be \( M_{1}(1 ; 0 ; 0), M_{2}(3 ; 1 ; 1) \). 1) Define the type of the surface \( \sigma \), 2) Draw the surface \( \sigma \), 3) Find sections of the surface \( \sigma \) with coordinate planes and define their foci and asymptotes, 4) Define the location of the points \( M_{1} \) and \( M_{2} \) in relation to the surface \( \sigma \) 5) Find the points of intersection of the straight line passing through points \( M_{1} \) and \( M_{2} \) with the surface \( \sigma \).

3.5.4 Surfaces of the 2-nd order

4.99 $