Problem 45
Question
Identify the quadric surface. $$ x^{2}-y+z^{2}=0 $$
Step-by-Step Solution
Verified Answer
The given equation represents a parabolic cylinder opening upwards along the y-axis.
1Step 1: Rearrange the terms
First, we rearrange the given equation \(x^{2}-y+z^{2}=0 \) to match the standard form of a quadric surface. This gives us \( x^{2} + z^{2} = y \)
2Step 2: Identify the quadric surface
The equation now looks like the general form of a parabolic cylinder, \(x^2 + z^2 = 4p(y-k) \), where p is the distance from the vertex to the focus, and k is the shift along the y-axis, with \(p=1/4\) and \(k=0\) in our case. It's important to note that a parabolic cylinder always opens in the direction of the isolated variable.
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