(a) Find and identify the traces of the quadric surface x 2 + y 2 z 2 = 4 given

Question

(a) Find and identify the traces of the quadric surface

x2 + y2 z2 = 4

given the plane.

x = k

Find the trace.


Identify the trace.

circle ellipse     hyperbola parabola

y = k

Find the trace.


Identify the trace.

circle ellipse     hyperbola parabola

z = k

Find the trace.


Identify the trace.

circle ellipse     hyperbola parabola

Describe the surface from one of the graphs in the table.

ellipsoid elliptic paraboloid     hyperbolic paraboloid cone hyperboloid of one sheet hyperboloid of two sheets

(b) If we change the equation in part (a) to

x2 y2 + z2 = 4,

how is the graph affected?

The graph is rotated so that its axis is the x-axis.

The graph is rotated so that its axis is the y-axis.   

The graph is rotated so that its axis is the z-axis

The graph is shifted one unit in the negative y-direction.

The graph is shifted one unit in the positive y-direction.

(c) What if we change the equation in part (a) to

x2 + y2 + 2y z2 = 3?

The graph is rotated so that its axis is the x-axis.

The graph is rotated so that its axis is the y-axis.    

The graph is rotated so that its axis is the z-axis

The graph is shifted one unit in the negative y-direction.

The graph is shifted one unit in the positive y-direction.

Explanation / Answer

(a) Given equation is of one sheeted hyperboloid having its axis along the z - axis

For z = k,Equation becomes

x2 + y2 = 4 + k2 which is a general equation of the circle hence represents a circle with origin as center and radius = (4 + k2)

For x = k,Equation becomes

For k 2

y2 - z2 = 4 - k2 which is a general equation of the hyperbola

For k = 2 it becomes

y2 - z2 = 0 which represents pair of straight line y = z and y = -z

For y = k,Equation becomes

For k 2

x2 - z2 = 4 - k2 which is a general equation of the hyperbola

For k = 2 it becomes

x2 - z2 = 0 which represents pair of straight line x = z and x = -z

(b) As the variable y and z get interchanged hence graph is rotated such that its axis is along the y-axis

(c) Rewriting the equation

=> x2 + y2 + 2y + 1 - z2 = 4

=> x2 + (y+1)2 - z2 = 4

=> x2 + (y - (-1))2 - z2 = 4

Comparing with the original equation, the graph is shifted toward y = -1 hence one unit in negative direction.

Note : Images are not visible in the problem

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