$\star$ Classify the surface
(a) $x^{2}+4 y^{2}+9 z^{2}=1$
(b) $y=2 x^{2}+z^{2}$
(c) $y^{2}=x^{2}+2 z^{2}$
(d) $x^{2}-6 x+4 y^{2}-z=0$
(e) $x^{2}+2 z^{2}=1$

Answer

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Answer

$x^{2}+2 z^{2}=1$ is an $\boxed{\text{elliptic cylinder}}$

Steps

Step 1 ：$x^{2}+4 y^{2}+9 z^{2}=1$ is an $\boxed{\text{ellipsoid}}$

Step 2 ：$y=2 x^{2}+z^{2}$ is an $\boxed{\text{paraboloid}}$

Step 3 ：$y^{2}=x^{2}+2 z^{2}$ is an $\boxed{\text{hyperboloid}}$

Step 4 ：$(x-3)^2 + 4y^2 - z = 9$ is an $\boxed{\text{elliptic paraboloid}}$

Step 5 ：$x^{2}+2 z^{2}=1$ is an $\boxed{\text{elliptic cylinder}}$