Question 1

Which of the following equations represents an elliptic paraboloid?

Accepted Answer

$z = x^2 + y^2$

Question 2

A surface defined by $x^2 + y^2 = 1$ is best described as:

Accepted Answer

A cylinder parallel to the $z$-axis

Question 3

Which quadric surface has horizontal traces that are ellipses and vertical traces that are hyperbolas?

Accepted Answer

Hyperboloid of one sheet

Question 4

Given the equation $x^2 + y^2 + z^2 = 9$, what is the geometric shape of the surface?

Accepted Answer

Sphere with radius 3

Question 5

Which of the following is the standard form of a quadric surface equation?

Accepted Answer

$Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0$

Question 6

If the equation of a surface is $y^2 + z^2 = 4$, which variable is missing and what does this imply about the surface?

Accepted Answer

The variable $x$ is missing; the surface is a cylinder parallel to the $x$-axis.

Question 7

Classify the surface given by $4x^2 - y^2 + 2z^2 + 4 = 0$.

Accepted Answer

Hyperboloid of two sheets

Question 8

Which surface is described by the equation $z^2 = x^2 + y^2$?

Accepted Answer

Cone

Question 9

A medical imaging device scans a region described by $\frac{x^2}{9} + \frac{y^2}{4} + \frac{z^2}{16} = 1$. What is the shape of the scanned region?

Accepted Answer

Ellipsoid

Question 10

Which of the following equations represents a hyperboloid of one sheet?

Accepted Answer

$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$

Question 11

A surface is defined by $z = x^2 - y^2$. What is the name of this surface?

Accepted Answer

Hyperbolic paraboloid

Question 12

Which quadric surface is used to model the shape of a satellite dish, given by $z = x^2 + y^2$?

Accepted Answer

Elliptic paraboloid

Question 13

If a surface is defined by $x^2 + y^2 - z^2 = 1$, what is the type of this surface?

Accepted Answer

Hyperboloid of one sheet

Question 14

Which of the following surfaces has traces in the $x$-$y$ plane that are ellipses and traces in the $x$-$z$ and $y$-$z$ planes that are parabolas?

Accepted Answer

Elliptic paraboloid

Question 15

A surface is described by $\frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$. What is the name of this surface?

Accepted Answer

Hyperboloid of two sheets

Question 16

Which surface is described by the equation $z = x^2 + y^2$ and is commonly used in the design of reflectors and antennas?

Accepted Answer

Elliptic paraboloid

Question 17

A surface is defined by $x^2 + y^2 = 4$. What is the geometric interpretation of this surface?

Accepted Answer

A circular cylinder parallel to the $z$-axis

Question 18

Which of the following equations represents a cone?

Accepted Answer

$z^2 = x^2 + y^2$

Question 19

A surface is defined by $z = x^2 - y^2$. What is the shape of the cross-section in the $x$-$z$ plane?

Accepted Answer

Parabola

Question 20

Which quadric surface is described by the equation $x^2 + y^2 + z^2 = r^2$?

Accepted Answer

Sphere

Section 2.6 - Cylinders and Quadric Surfaces

Study Guide - Practice Questions