Which surface is described by x²/4 + y²/9 - z² = 1?
AEllipsoid
BHyperboloid of one sheet
CHyperboloid of two sheets
DElliptic cone
Two squared terms are positive and one is negative, with 1 on the right side. The traces z = k are ellipses x²/4 + y²/9 = 1 + k², which exist for every k, so the surface is a single connected piece: a hyperboloid of one sheet.
Question 2 True / False
In three-dimensional space, the equation x² + y² = 4 describes a circle of radius 2.
TTrue
FFalse
Answer: False
Because z does not appear, the equation places no restriction on z. Every horizontal plane z = k contains the same circle, so the graph is a circular cylinder of radius 2 around the z-axis.
Question 3 Short Answer
Describe the traces of z = x² - y² in the planes y = 0, x = 0 and z = 1, and name the surface.
Think about your answer, then reveal below.
Model answer: y = 0 gives the upward parabola z = x²; x = 0 gives the downward parabola z = -y²; z = 1 gives the hyperbola x² - y² = 1. The surface is a hyperbolic paraboloid (saddle).
Opposite-facing parabolas in perpendicular directions are the signature of a saddle. Horizontal traces are hyperbolas, except at z = 0, where the trace is the pair of lines y = ±x.
Question 4 Multiple Choice
Which equation describes an elliptic paraboloid opening along the positive y-axis?
Ay = x² + z²
By² = x² + z²
Cx² + y² + z² = 1
Dy = x² - z²
The variable that appears to the first power marks the axis of the paraboloid. In y = x² + z², the traces y = k > 0 are circles and y ≥ 0 everywhere, so it opens toward positive y. y² = x² + z² is a cone and y = x² - z² is a saddle.