Question 1

Which of the following equations represents an elliptic cone?

Accepted Answer

$$\frac{x^2$}{a^2$} + \frac{y^2$}{b^2$} - \frac{z^2$}{c^2$} = 0$$

Question 2

For the surface $$z = \frac{x^2$}{a^2$} + \frac{y^2$}{b^2$}$$, what is the shape of the trace in the plane $$z = p ($$where $$p is a $$constant)?

Accepted Answer

An ellipse

Question 3

Which quadric surface has traces that are parabolas in both the $$x = r$$ and $$y = q$$ planes, and an ellipse in the $$z = p$$ plane?

Accepted Answer

Elliptic paraboloid

Question 4

In the equation $$z = \frac{x^2$}{a^2$} - \frac{y^2$}{b^2$}$$, what is the shape of the trace in the plane $$z = p$$?

Accepted Answer

A hyperbola

Question 5

For the elliptic cone $$\frac{x^2$}{a^2$} + \frac{y^2$}{b^2$} - \frac{z^2$}{c^2$} = 0$$, which variable corresponds to the axis of the surface?

Accepted Answer

$$z$$

Question 6

A satellite dish is often shaped like a paraboloid. If its cross-section in the $$z = p$$ plane is an ellipse, which equation best models its surface?

Accepted Answer

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

Question 7

Which of the following statements is true about the traces of a hyperbolic paraboloid $$z = \frac{x^2$}{a^2$} - \frac{y^2$}{b^2$}$$?

Accepted Answer

Traces in $$z = p$$ planes are hyperbolas

Question 8

If you slice an elliptic cone with the plane $$z = p$$, what type of curve do you obtain?

Accepted Answer

An ellipse

Question 9

Which surface has traces that are a pair of lines intersecting at the origin in both the $$xz$$ and $$yz$$ planes?

Accepted Answer

Elliptic cone

Question 10

A bridge designer wants a surface whose cross-sections in the $$y = q$$ and $$x = r$$ planes are parabolas, but in the $$z = p$$ plane is a hyperbola. Which surface should they use?

Accepted Answer

Hyperbolic paraboloid

Question 11

For the surface $$z = \frac{x^2$}{a^2$} + \frac{y^2$}{b^2$}$$, which variable is the axis of the surface?

Accepted Answer

$$z$$

Question 12

Which of the following best describes the traces of an elliptic cone in the $$y = q$$ and $$x = r$$ planes?

Accepted Answer

Both are hyperbolas

Question 13

A surface is defined by $$z = \frac{x^2$}{a^2$} - \frac{y^2$}{b^2$}. $$What is the shape of the trace in the $$y = q$$ plane?

Accepted Answer

A parabola

Question 14

Which of the following surfaces has an axis corresponding to the variable with a negative coefficient?

Accepted Answer

Elliptic cone

Question 15

If a surface has traces that are parabolas in both $$x = r$$ and $$y = q$$ planes, and a hyperbola in the $$z = p$$ plane, which equation represents it?

Accepted Answer

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

Question 16

Which of the following is NOT a characteristic of the elliptic cone $$\frac{x^2$}{a^2$} + \frac{y^2$}{b^2$} - \frac{z^2$}{c^2$} = 0$$?

Accepted Answer

Traces in $$z = p$$ planes are parabolas

Question 17

A real-world application of a hyperbolic paraboloid is the design of certain roofs. Which property of the surface makes it suitable for this purpose?

Accepted Answer

It is a ruled surface, allowing straight beams to form a curved roof

Question 18

Which surface would you use to model a flashlight reflector, given that its cross-sections perpendicular to the axis are ellipses and along the axis are parabolas?

Accepted Answer

Elliptic paraboloid

Question 19

If the equation of a surface is $$z = \frac{x^2$}{a^2$} + \frac{y^2$}{b^2$}$$, what is the shape of the trace in the $$x = r$$ plane?

Accepted Answer

A parabola

Question 20

Which of the following best describes the axis of the hyperbolic paraboloid $$z = \frac{x^2$}{a^2$} - \frac{y^2$}{b^2$}$$?

Accepted Answer

The variable that appears linearly

Characteristics of Common Quadric Surfaces

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