Check whether the following equations.... (analytical geometry)

Question

Check whether the following equations represent paraboloids or not. For those, that do, classify the paraboloids as elliptic or hyperbolic... (i). x^2 - y^2 + 2x + 7y + z + 4 = 0.... (ii). 2(x^2) + z^2 = 4..... (iii). 2(x^2) + 3(y^2) + z^2 - x - y = 1...... (iv). x^2 = 2(y + z)

Answer 1

Briefly:

(i) looks like a hyperboloid, but there is an extraneous z term. This implies that when viewed along the z axis, we have an off-centre hyperbola (x^2 and y^2 terms with a displacement of the centre) as a cross-section. This cross-section has varying parameters depending on z.

(ii) is an ellipse or a cylindrical ellipse (fixed cross-section).

(iii) is an ellipsoid.

(iv) is neither an ellipsoid nor a hyperboloid.

I hope this is meaningful.