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 #46164 – Math – Analytic Geometry

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

Solution.

$$f(x, y, z) = x^2 + y^2 + z^2 - 2x - 4z + 1,$$ $$2x + y + z = 2 \rightarrow z = 2 - 2x - y.$$ $$\begin{array}{l} \text{So, } f(x, y) = x^2 + y^2 + (2 - 2x - y)^2 - 2x - 4(2 - 2x - y) + 1 = \\ = 5x^2 + 2y^2 + 4xy - 2x - 3. \end{array}$$

Therefore, equation of the cylinder is

$$5x^2 + 2y^2 + 4xy - 2x - 3 = 0.$$

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