Question

Find the equation of the cylinder with base curve x^2+y^2+z^2-2x-4z+1=0,2x+y+z=2.

Answer 1

Answer on Question #45081 – Math – Analytic Geometry

Task:

Find the equation of the cylinder with base curve

$$x ^ {2} + y ^ {2} + z ^ {2} - 2 x - 4 z + 1 = 0, \ 2 x + y + z = 2$$

Solution:

The first equation is the sphere. Indeed,

$$x ^ {2} - 2 x + 1 - 1 + y ^ {2} + z ^ {2} - 4 z + 4 - 4 + 1 = 0$$ $$(x - 1) ^ {2} + y ^ {2} + (z - 2) ^ {2} = 4$$

And the last one is the plane.

We can find the equation of the cylinder by the substitution. From the equation

Second equation we have $z = 2 - 2x - y$ . And from the second equation we get

$$x ^ {2} + y ^ {2} + (2 - 2 x - y) ^ {2} - 2 x - 4 (2 - 2 x - y) + 1 = 0$$ $$x ^ {2} + y ^ {2} + 4 + 4 x ^ {2} + y ^ {2} - 8 x - 4 y + 4 x y - 2 x - 8 + 8 x + 4 y + 1 = 0$$ $$5 x ^ {2} + 2 y ^ {2} + 4 x y - 2 x - 3 = 0$$

Answer: $5x^{2} + 2y^{2} + 4xy - 2x - 3 = 0$

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