a.) Find and identify the traces of the quadric surfacex^2+y^2-z^2= 1 and use the traces to classify the graph of the surface.
b.) If we change the equation in part a.) to x^2-y^2+z^2= 1, how is the graph affected?
c.) What if we change the equation in part a.) to x^2+y^2+ 2y-z^2= 0? How is the graph affected in this case?
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Expert's answer
2020-09-14T19:20:27-0400
a) x2+y2−z2=1
x=k=>k2+y2−z2=1=>y2−z2=1−k2
The trace is a hyperbola when k=±1.
If k=±1,y2−z2=(y−z)(y+z)=0, so it is a union of two lines.
y=k=>x2+k2−z2=1=>x2−z2=1−k2
The trace is a hyperbola when k=±1.
If k=±1,x2−z2=(x−z)(x+z)=0, so it is a union of two lines.
z=k=>x2+y2−k2=1=>x2+y2=1+k2
The trace is a circle whose radius is 1+k2.
Therefore the surface is a stack of circles, whose traces of other directions are
hyperbolas. So it is a hyperboloid. The intersection with the plane z=k is
never empty. This implies the hyperboloid is connected.
b) The role of y and z are interchanged. So now the axis of given hyperboloid
is y -axis.
c)
x2+y2+2y−z2=0=>
x2+(y2+2y+1)−z2=1=>
x2+(y+1)2−z2=1
Thus it is a translation of the hyperboloid x2+y2−z2=1 by (0,−1,0).
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