Concepts and Vocabulary
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) ________ ________.
Skill Building
In Problems 7–18:
\(z=x^{2}+y^{2}\)
\(z=x^{2}-y^{2}\)
\(4x^{2}+y^{2}+4z^{2}=4\)
\(2x^{2}+y^{2}+z^{2}=1\)
\(z^{2}=x^{2}+2y^{2}\)
\(x^{2}+2y^{2}-z^{2}=1\)
\(x=4z^{2}\)
\(x^{2}+y^{2}=1\)
\(x^{2}+2y^{2}-z^{2}=-4\)
\(y^{2}-x^{2}=4\)
\(2x=y^{2}\)
\(4y^{2}-x^{2}=1\)
In Problems 19–24, use the Figures A-F to match each graph to an equation.
\(z=4y^{2}-x^{2}\)
\(2z=x^{2}+4y^{2}\)
\(2x^{2}+y^{2}-z^{2}=1\)
\(2x^{2}+y^{2}+3z^{2}=1\)
\(y^{2}=4x\)
\(x^{2}-z^{2}=y\)
752
Figures G–L are graphs of quadric surfaces. In Problems 25–30, match each equation with its graph.
\(3x^{2}+4y^{2}+z=0\)
\(3x^{2}+4y^{2}+4y=0\)
\(3x^{2}+2y^{2}-( {z-2}) ^{2}+1=0\)
\(z^{2}-4x^{2}=3y\)
\(x^{2}+2y^{2}-z^{2}+4z=4\)
\(3x^{2}+3y^{2}+z^{2}=1\)
Applications and Extensions
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:
Challenge Problem
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.)