10.7 Assess Your Understanding

Concepts and Vocabulary

Question

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) \(\dfrac{x^2}{2}-\dfrac{y^2}{3}=1\bigg]\).

Question

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

Question

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

Question

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].

Question

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

Question

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:

  1. Identify the equation of each quadric surface.
  2. List the intercepts and traces.
  3. Graph each quadric surface.

Question

\(z=x^{2}+y^{2}\)

Question

\(z=x^{2}-y^{2}\)

Question

\(4x^{2}+y^{2}+4z^{2}=4\)

Question

\(2x^{2}+y^{2}+z^{2}=1\)

Question

\(z^{2}=x^{2}+2y^{2}\)

Question

\(x^{2}+2y^{2}-z^{2}=1\)

Question

\(x=4z^{2}\)

Question

\(x^{2}+y^{2}=1\)

Question

\(x^{2}+2y^{2}-z^{2}=-4\)

Question

\(y^{2}-x^{2}=4\)

Question

\(2x=y^{2}\)

Question

\(4y^{2}-x^{2}=1\)

In Problems 19–24, use the Figures A-F to match each graph to an equation.

Question

\(z=4y^{2}-x^{2}\)

Question

\(2z=x^{2}+4y^{2}\)

Question

\(2x^{2}+y^{2}-z^{2}=1\)

Question

\(2x^{2}+y^{2}+3z^{2}=1\)

Question

\(y^{2}=4x\)

Question

\(x^{2}-z^{2}=y\)

752

Figures G–L are graphs of quadric surfaces. In Problems 25–30, match each equation with its graph.

Question

\(3x^{2}+4y^{2}+z=0\)

Question

\(3x^{2}+4y^{2}+4y=0\)

Question

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

Question

\(z^{2}-4x^{2}=3y\)

Question

\(x^{2}+2y^{2}-z^{2}+4z=4\)

Question

\(3x^{2}+3y^{2}+z^{2}=1\)

Applications and Extensions

Question

Explain why the graph of \(xy=1\) in space is a cylinder.

Question

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

Question

Graph:

  1. \(xy=1\)
  2. \(z=\sin y\)
  3. \(z=xy\). (This surface is a hyperbolic paraboloid rotated \(45^\circ\) about the \(z\)-axis.)

Challenge Problem

Question

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.)