course Mth 272
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11:38:01
Query problem 7.2.52 (was 7.2.48) identify quadric surface z^2 = x^2 + y^2/2.
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RESPONSE -->
confidence assessment:
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11:38:20
What is the name of this quadric surface, and why?
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RESPONSE -->
Eliptic cone
x^2 + y^2/4 - x^2 = 0
confidence assessment: 2
That is correct. The reasons:
** If z = c, a constant, then x^2 + y^2/2 = c^2, or x^2 / c^2 + y^2 / (`sqrt(2) * c)^2 = 1. This gives you ellipse with major axis c and minor axis `sqrt(2) * c. Thus for any plane parallel to the x-y plane and lying at distance c from the x-y plane, the trace of the surface is an ellipse.
In the x-z plane the trace is x^2 - z^2 = 0, or x^2 = z^2, or x = +- z. Thus the trace in the x-z plane is two straight lines.
In the y-z plane the trace is y^2 - z^2/2 = 0, or y^2 = z^2/2, or y = +- z * `sqrt(2) / 2. Thus the trace in the y-z plane is two straight lines.
The x-z and y-z traces show you that the ellipses in the 'horizontal' planes change linearly with their distance from the x-y plane. This is the way cones grow, with straight lines running up and down from the apex. Thus the surface is an elliptical cone. **
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11:38:31
Give the equation of the xz trace of this surface and describe its shape, including a justification for your answer.
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RESPONSE -->
To find the xz trace make y = 0
x^2 - z^2 = 0
confidence assessment: 2
** The xz trace consists of the y = 0 points, which for z^2 = x^2 + y^2/2 is z^2 = x^2 + 0^2/2 or just z^2 = x^2.
The graph of z^2 = x^2 consists of the two lines z = x and z = -x in the yz plane. **
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11:38:42
Describe in detail the z = 2 trace of this surface.
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RESPONSE -->
x^2 - 2^2 = 0
x^2 = 4
x = 2
The trace will intersect the x axis at 2 and the z axis at 2
confidence assessment: 2
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** If z = 2 then z^2 = x^2 + y^2/2 becomes 2^2 = x^2 + y^2 / 2, or x^2 + y^2 / 2 = 4.
This is an ellipse. If we divide both sides by 4 we can get the standard form:
x^2 / 4 + y^2 / 8 = 1, or x^2 / 2^2 + y^2 / (2 `sqrt(2))^2 = 1.
This is an ellipse with major axis 2 `sqrt(2) in the y direction and 2 in the x direction. **
&#Good work. See my notes and let me know if you have questions. &#