Consider the surface given by the equation:
2x2 - 3z2 - 4x -4y - 12z + 2 = 0
(a) Write in standard form )
(b) Determine and classify the traces with planes parallel to the xy-, xz- and yz- planes.
(c) Hence classify the surface.
2x^2 - 3z^2 - 4x - 4y - 12z + 2 = 0
2(x-1)^2-2 - 3(z+2)^2+12 - 4y + 2 = 0
2(x-1)^2 - 3(x+2)^2 - 4y + 12 = 0
4(y-3) = 2(x-1)^2 - 3(z+2)^2
Looks like an hyperbolic paraboloid with vertex at (1,3,-2)
The traces are an hyperbola in the xz plane, and parabolas in the others.
(a) To write the given equation in standard form, we need to rearrange the terms so that the variables are grouped together and the constant term is on the other side of the equation.
Rearranging the terms, we have:
2x^2 - 3z^2 - 4x - 4y - 12z + 2 = 0
Grouping the x, y, and z terms:
2x^2 - 4x - 3z^2 - 12z - 4y + 2 = 0
Write the x terms:
2x^2 - 4x
Write the z terms:
-3z^2 - 12z
Write the y term:
-4y
Now we have:
2x^2 - 4x - 3z^2 - 12z - 4y = -2
(b) To determine and classify the traces with planes parallel to the xy-, xz-, and yz-planes, we will assign fixed values to one variable at a time and find the corresponding equation for the resulting two-variable surface.
(i) Traces with planes parallel to the xy-plane (z = constant):
Setting z = k (where k is a constant), we can rewrite the equation as:
2x^2 - 4x - 3k^2 - 12k - 4y = -2
This is a quadratic equation in x and y. By solving this equation, we can obtain the equation of the trace of the surface on the xy-plane.
(ii) Traces with planes parallel to the xz-plane (y = constant):
Setting y = k (where k is a constant), we can rewrite the equation as:
2x^2 - 4x - 3z^2 - 12z - 4k = -2
This is a quadratic equation in x and z. By solving this equation, we can obtain the equation of the trace of the surface on the xz-plane.
(iii) Traces with planes parallel to the yz-plane (x = constant):
Setting x = k (where k is a constant), we can rewrite the equation as:
2k^2 - 4k - 3z^2 - 12z - 4y = -2
This is a quadratic equation in y and z. By solving this equation, we can obtain the equation of the trace of the surface on the yz-plane.
(c) Once we have determined the equations of the traces on the xy-, xz-, and yz-planes, we can use their forms to classify the surface.
By analyzing the equations of the traces, we can determine the shape and properties of the surface. Common surface classifications include sphere, ellipsoid, cone, hyperboloid, and so on. To classify the surface, we need to examine these equations and identify the corresponding geometric shape.