Question

x^2+y^2=a^2z^2 represents

Answer 1

We first set y and z equal to 0 to get the point

(6,0,0)

Similarly, we find the other two intercepts

(0,4,0) and (0,0,3)

Now plot the three points and connect the dots as shown in the picture below.

Quadric Surfaces

In the xy-plane the next step after studying lines is the study of conics: parabolas, ellipses, and hyperbolae. Their equations all have x2 or y2 terms or both. In three dimensions surfaces whose equations have only linear and quadratic terms are called quadric surfaces. The naming devise uses the suffix "-oid" to indicate that the surfaces has a trace in the shape of an ellipse. Note that a circle is a special ellipse. Below are names of some of these:

x2/a2 + y2/b2 + z2/c2 = 1 is an ellipsoid

-x2/a2 - y2/b2 + z2/c2 = 1 is a hyperboloid of 2 sheets while

x2/a2 + y2/b2 - z2/c2 = 1 is a hyperboloid of 1 sheet

z = x2/a2 + y2/b2 is a paraboloid

z = x2/a2 - y2/b2 is a hyperbolic paraboloid

x2/a2 + y2/b2 - z2/c2 = 0 is a cone

Example

Name the following quadric

Solution

Notice that the trace on the xy-plane is

x2 - y2 = 1

is a hyperbola and on the xz-plane is

x2 - 4z2 = 1

is also a hyperbola and no the yz-plane is

x2 + 4z2 = 1

is an ellipse. Since there is only one negative, we see that the surface is a hyperboloid of one sheet. Its axis is the y-axis.

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