Multiple Choice The trace in the $xy$-plane of the graph of the equation $\dfrac{ x^{2}}{2}-\dfrac{y^{2}}{3}+\dfrac{z^{2}}{4}=1$ is $\bigg[$(a) $\dfrac{x^2}{2} + \dfrac{z^2}{4}=1,$ (b) $\dfrac{x^2}{2}-\dfrac{y^2}{3}=0,$ (c) $\dfrac{x^2}{2}-\dfrac{y^2}{3}=1\bigg]$.

The intercept(s) of the graph of $\dfrac{x^{2}}{4}-\dfrac{y^{2}}{ 9}-z=4$ is(are) ________.

True or False A cylinder is formed when a line moves along a plane curve while remaining perpendicular to the plane containing the curve.

Multiple Choice The quadric surface $z^{2}=x^{2}+\dfrac{y^{2}}{4}$ is called a(n) [(a) elliptic cylinder, (b) elliptic cone, (c) elliptic paraboloid, (d) hyperboloid].

The quadric surface $y^{2}-x^{2}=4$ is called a(n) ________.

The point $( 0,0,0)$ on the hyperbolic paraboloid $z= \dfrac{y^{2}}{2^{2}}-\dfrac{x^{2}}{5^{2}}$ is called a(n) ________ ________.

$z=x^{2}+y^{2}$

$z=x^{2}-y^{2}$

$4x^{2}+y^{2}+4z^{2}=4$

$2x^{2}+y^{2}+z^{2}=1$

$z^{2}=x^{2}+2y^{2}$

$x^{2}+2y^{2}-z^{2}=1$

$x=4z^{2}$

$x^{2}+y^{2}=1$

$x^{2}+2y^{2}-z^{2}=-4$

$y^{2}-x^{2}=4$

$2x=y^{2}$

$4y^{2}-x^{2}=1$

$z=4y^{2}-x^{2}$

$2z=x^{2}+4y^{2}$

$2x^{2}+y^{2}-z^{2}=1$

$2x^{2}+y^{2}+3z^{2}=1$

$y^{2}=4x$

$x^{2}-z^{2}=y$

$3x^{2}+4y^{2}+z=0$

$3x^{2}+4y^{2}+4y=0$

$3x^{2}+2y^{2}-( {z-2}) ^{2}+1=0$

$z^{2}-4x^{2}=3y$

$x^{2}+2y^{2}-z^{2}+4z=4$

$3x^{2}+3y^{2}+z^{2}=1$

Explain why the graph of $xy=1$ in space is a cylinder.

Explain why the graph of $z=\sin y$ in space is a cylinder.

Graph:

Show that through each point on the hyperboloid of one sheet $$\begin{equation*} \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1 \end{equation*}$$

there are two lines lying entirely on the surface. (Hint: Write the equation as $\dfrac{x^{2}}{a^{2}}-\dfrac{z^{2}}{c^{2}}=1-\dfrac{y^{2}}{ b^{2}}$ and factor.)