Answer
Hyperboloid of One Sheet directed along \(x\)-axis.
Work Step by Step
We can rearrange the terms for it to be in standard form:
\[
9y^2+4z^2=x^2+36 \\
-x^2+9y^2+4z^2=36 \\
-\frac{x^2}{36}+\frac{9y^2}{36}+\frac{4z^2}{36}=1 \\
\]
Evaluated at \(x=0\):
\[
-\frac{(0)^2}{36}+\frac{9y^2}{36}+\frac{4z^2}{36}=1 \\
\frac{y^2}{4}+\frac{z^2}{9}=1 \\
\frac{y^2}{(2)^2}+\frac{z^2}{(3)^2}=1 \\
\]
Ellipse of radius \(2\) along the \(y\)-axis and \(3\) along the \(z\)-axis.
Evaluated at \(x=\pm 2\):
\[
-\frac{(\pm6)^2}{36}+\frac{9y^2}{36}+\frac{4z^2}{36}=1 \\
-1+\frac{y^2}{4}+\frac{z^2}{9}=1 \\
\frac{y^2}{4}+\frac{z^2}{9}=2 \\
\frac{y^2}{8}+\frac{z^2}{18}=1 \\
\frac{y^2}{2\sqrt{2}}+\frac{z^2}{3\sqrt{2}}=1 \\
\]
Ellipse of radius \(2\sqrt{2}\) along the \(y\)-axis and \(3\sqrt{2}\) along the \(z\)-axis.
