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 \(x = \pm 6\):
\[
-\frac{(\pm 6)^2}{36} + \frac{9y^2}{36} + \frac{4z^2}{36} = 1 \\
-1 + \frac{y^2}{4} + \frac{z^2}{9} = 1 \\
\frac{y^2}{8} + \frac{z^2}{18} = 1 \\
\frac{y^2}{(2\sqrt{2})^2} + \frac{z^2}{(3\sqrt{2})^2} = 1
\]
Ellipse of radius \(2\sqrt{2}\) along the \(y\)-axis and \(3\sqrt{2}\) along the \(z\)-axis.
