(a) Find and identify the traces of the quadric surface x2 + y2 − z2 =...

Question

Question

(a) Find and identify the traces of the quadric surface x2 + y2 − z2 =...

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

x² + y² − z² = 16

given the plane.

x = k

Find the trace.

Identify the trace.

circleellipse hyperbolaparabola

y = k

Find the trace.

Identify the trace.

circleellipse hyperbolaparabola

z = k

Find the trace.

Identify the trace.

circleellipse hyperbolaparabola

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

ellipsoidelliptic paraboloid hyperbolic paraboloidconehyperboloid of one sheethyperboloid of two sheets

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

x² − y² + z² = 16,

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-axisThe 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 = 15?

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

math calculus

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Answer 1

Answer #1

x^2 + y^2 - z^2 = 16
x = k

k + y^2 - z^2 = 16

y^2 - z^2= 16 - k^2

y^2 - z^2 = c, where c = 16 - k^2

So, this is a HYPERBOLA with major axis on
the y-axis and center = 0,0

hyperbolic paraboloid

------------------------------------------------

y = k :

x^2 - z^2 = c

So, this is a HYPERBOLA in xz plane
with center = 0,0

hyperbolic paraboloid

-----------------------------------------------

z = k :
x^2 + y^2 = 16 + k^2

So, circle on xy plane
with radius = sqrt(16 + k^2)

Elliptic Paraboloid

------------------------------------------------

b)
x^2 - y^2+ z^2 = 16

x = k :
z^2 - y^2 = 16 - k^2
HYPERBOLA

y = k :
x^2 + z^2= 16 + k^2
CIRCLE

z = k :
x^2 - y^2 = 16 - k^2
HYPERBOLA

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

----------------------------------------------------------------------------------------------------------------------

c)

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

Add a comment

Answer 2

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