Problem 52
Question
Identify the quadric surface. $$ 4 y=x^{2}+z^{2} $$
Step-by-Step Solution
Verified Answer
The given quadric surface is an upward-opening paraboloid.
1Step 1: Match the equation with the standard form of the quadric surfaces
First, recognize the given equation in the standard form of quadric surfaces. The given equation is \(4y = x^{2} + z^{2}\). This can be rearranged to \(y = 1/4x^{2} + 1/4z^{2}\) which matches the standard form of a paraboloid: \(y = ax^{2} + bz^{2}\) where \(a = 1/4\) and \(b = 1/4\). Note that here 'a' and 'b' are positive constants.
2Step 2: Identify the type of paraboloid
The signs of 'a' and 'b' in the standard form of a paraboloid help us to determine whether the paraboloid opens upwards or downwards. In this case, as both 'a' and 'b' are positive, the paraboloid opens upwards.
3Step 3: Conclusion
Putting it all together, the given equation \(4y = x^{2} + z^{2}\) represents an upward-opening paraboloid.
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