Question 1

Which of the following is the general equation for a quadric surface in three variables?

Accepted Answer

$$A x^2$ + B y^2$ + C z^2$ + D x y + E x z + F y z + G x + H y + I z + J = 0$$

Question 2

A quadric surface has hyperbolas as its $$xy-$$trace and $$yz-$$trace, and for each $$y = y_0$ its trace is an ellipse. What type of surface is this?

Accepted Answer

A hyperboloid of two sheets

Question 3

Given the cone $$x^2$$ + $$y^2$$ = $$z^2$$, what is the $$x = x_0$ trace (for x_0$ 
eq 0$$)?

Accepted Answer

A hyperbola

Question 4

What type of quadric surface is represented by the equation $$x^2 + 2x$$ + $$y^2$$ = $$z^2$$?

Accepted Answer

A hyperboloid of one sheet centered at $$(-1, 0, 0)$$

Question 5

What is the $$x = 1$$ trace of the surface $$x^2 - 2x$$ + $$y^2$$ = $$z^2 + 2z$?

Accepted Answer

A pair of lines

Question 6

Which of the following is NOT a possible trace of an ellipsoid?

Accepted Answer

A hyperbola

Question 7

Which type of quadric surface cannot have empty traces?

Accepted Answer

Hyperboloid of one sheet

Question 8

What is the equation for the surface obtained by revolving the ellipse $$x^2$$ + $$4y^2 = 4$ in the xy$-plane about the x$-axis?

Accepted Answer

$$x^2$$ + $$4y^2$$ + $$4z^2 = 4$

Question 9

Suppose you have the surface $$x^2$$ + $$y^2$$ - $$z^2 = 1$. What is the yz$-trace when x = 0$?

Accepted Answer

$$y^2$$ - $$z^2 = 1$

Question 10

Which of the following surfaces can have a trace that is a single point?

Accepted Answer

Ellipsoid

Question 11

A surface has the equation $$(x+1)^2$$ + $$y^2$$ - $$z^2 = 1$. What is the center of this surface?

Accepted Answer

$$(-1, 0, 0)$$

Question 12

Which of the following best describes the $$y = y_0$ trace of the cone x^2$ + y^2$ = z^2$ for y_0$ 
eq 0$$?

Accepted Answer

A hyperbola opening along the $$z-$$axis

Question 13

Which of the following quadric surfaces is always centered at the origin unless shifted by completing the square?

Accepted Answer

All of the above, unless shifted

Question 14

If a quadric surface has the equation $$x^2$$ + $$y^2$$ - $$z^2 = 0$, what is the x = 0$ trace?

Accepted Answer

$$y^2$$ - $$z^2 = 0$

Question 15

Which of the following surfaces could have a trace that is empty (no real solution)?

Accepted Answer

Ellipsoid

Question 16

A medical imaging device scans a cross-section of a patient's body and finds the trace is a pair of lines. Which quadric surface could model this cross-section?

Accepted Answer

A cone

Question 17

Which of the following equations represents a surface whose $$yz-$$trace is always a circle (for all $$x$$)?

Accepted Answer

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

Question 18

A chemical reaction chamber is modeled by the surface $$x^2$$ + $$4y^2$$ + $$4z^2 = 4$. What is the shape of the cross-section when x = 0$?

Accepted Answer

An ellipse

Question 19

Which of the following surfaces can have traces that are both ellipses and points?

Accepted Answer

Ellipsoid

Question 20

A surface has the equation $$x^2$$ + $$y^2$$ = $$z^2. $$What is the $$z = 0$$ trace?

Accepted Answer

$$x^2$$ + $$y^2 = 0$

Quadric Surfaces and Their Traces: Study Notes for Calculus Students

Study Guide - Practice Questions

Study Guide - Flashcards