Chapter 3: Functions of Several Variables
Section 3.3: Quadric Surfaces
Example 3.3.7
Put the equation 9x2+4y2−18 x+16 y−11=0 into standard form for a quadric surface, identify the surface, draw its graph, and discuss the nature of the level curves and plane sections.
Solution
Mathematical Solution
Figures 3.3.7(a, b) contain a graph of the surface defined by the given equation,
9x2+4y2−18 x+16 y−11=0
whose standard form is
x−1222+y+2232=1
obtained by completing the square in x and y. The standard form is the equation of an elliptic cylinder.
The point 1,−2 would be the center of the ellipse in the xy-plane.
The level curves, drawn on the surface of the cylinder, are ellipses with semi-major axis 3 and semi-minor axis 2.
The cross sections x=c are vertical lines y=−2 ±33+2 c−c2/2 in the yz-plane.
The cross sections y=c are pairs of vertical lines x=1 ±25−4 c−c2/3 in the xz-plane.
x = =
Figure 3.3.7(a) Elliptic cylinder with cross sections x=c
y = =
Figure 3.3.7(b) Elliptic cylinder with cross sections y=c
Maple Solution - Interactive
Obtain the standard form
Control-drag the given equation.
Context Panel: Manipulate Equation
Check the "Show steps stacked vertically" box.
Click the "Complete the square" button.
Add to both sides and multiply both sides as per the actions shown in the figure below.
Click the "Return Steps" button.
9x2+4y2−18 x+16 y−11=0→manipulate equation19y+22+14x−12=1
Graph via the Context Panel
Control-drag the equation.
Context Panel: Plots≻3-D ImplicitPlot≻x, y, ? Style≻Surface Contour
9x2+4y2−18 x+16 y−11=0→
Alternative, use the to obtain the equivalent of the surfaces in Figures 3.3.7(a, b)
Click the "Interactive Plot Builder" button.
On the main panel, click "Edit Functions", thereby launching the "Specify Expressions" dialog. In the Variables section, click Add, and enter z as an additional variable. See Figure 3.3.7(d). After clicking the Add button, the bottom pane will be empty. Click OK.
Set the ranges −1≤x≤3,−5≤y≤1,0≤z≤5 as per Figure 3.3.7(c).
Press the Options button. Set the options as per Figure 3.3.7(f).
Press the Advanced Settings button in the Axes section; adjust the tickmarks as per Figure 3.3.7(e).
Figure 3.3.7(c) Main panel of Interactive Plot Builder
Figure 3.3.7(d) Settings for tickmarks
Figure 3.3.7(e) Settings for tickmarks
Figure 3.3.7(f) Options panel, Interactive Plot Builder
Maple Solution - Coded
Define f so that the graph of f=0 is a quadric surface
f≔9x2+4y2−18 x+16 y−11:
Complete the square and put f into standard form
Student:-Precalculus:-CompleteSquaref=0,x,y
4y+22+9x−12−36=0
−−36
4y+22+9x−12=36
/36
y+229+x−124=1
Obtain the equivalent the surface in Figures 3.3.7(a, b)
plots:-implicitplot3df=0,x=−1..3,y=−5..1,z=0..5,scaling=constrained,style=surfacecontour,axes=frame,orientation=−50,75,0,tickmarks=3,4,5
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