Question

Identify the quadric surface. $$ 4 y=x^{2}+z^{2} $$

Short answer

The given quadric surface is an upward-opening paraboloid.

Step-by-step solution

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.

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.

Conclusion

Putting it all together, the given equation \(4y = x^{2} + z^{2}\) represents an upward-opening paraboloid.