about quadric surfaces?

Question

please help i need it badly....i need to understand this T_T

1. z=y^2/25-x^2,this surface is hyperbolic surface,what is the opening of this equation(i need the pening to determine the parallel) and the traces and the parallel to these..

2. x^2+y^2-z^2= 16,what is the opening and the traces...i'm not sure if the equation is a ellipsoid.....


please help me ......many thanks

Answer 1

what do you mean by opening....
as for the traces,
you set
z= constants, for example, with x^2+y^2-z^2= 16
you would get:
z=0, x^2+y^2= 16 which is a circle of radius 4
z=1,-1, x^2+y^2-1^2= 16, which is a circle of radios sqrt(17)
z=2,-2, x^2+y^2-2^2= 16, which is a circle of radius sqrt(12)

x= constants:,
x=0 y^2-z^2= 16, which is a hyperbola in the yz plane
x=1 1^2+y^2-z^2= 16, y^2-z^2= 15, hyperbola
x=2 y^2-z^2 =12, etc etc

y=constants:
y=0, x^2-z^2= 16 hyperbola in the xz plane
y=1 x^2 - z^2 =15, etcetcetc

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Answer 2

1. The hyperbola opens up in the y-z plane, and down in the x-z plane. I'm not sure what you mean by traces, but in the y-z plane the curve will be z = y^2/25, and in the x-z plane, z = -x^2

2) x^2+y^2-z^2= 16 is a hyperboloid of revolution about the z- axis, and opens to ± z. Traces are:
x^2 + y^2 = 16
z^2 = x^2 - 16, x ≥ 4
Z^2 = y^2 - 16, y ≥ 4