2. The traces in the coordinate planes refer to the curves you get when a surface intersects one of the three coordinate planes in $$ 3 d $$. What are the traces in the coordinate planes for the following surfaces? Sketch them on a single $$ 3 d $$ space. Can you name the surfaces? (a) $$ 2 x+4 y+3 z=12 $$ (b) $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1 $$

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**(a)**  
We are given the plane  
$$
2x + 4y + 3z = 12.
$$

• In the $$xy$$-plane set $$z=0$$. Then the equation becomes  
$$
2x + 4y = 12 \quad \Longrightarrow \quad x + 2y = 6.
$$
This is a line in the $$xy$$-plane.

• In the $$xz$$-plane set $$y=0$$. Then the equation becomes  
$$
2x + 3z = 12 \quad \Longrightarrow \quad z = \frac{12 - 2x}{3}.
$$
This is a line in the $$xz$$-plane.

• In the $$yz$$-plane set $$x=0$$. Then the equation becomes  
$$
4y + 3z = 12 \quad \Longrightarrow \quad z = \frac{12 - 4y}{3}.
$$
This is a line in the $$yz$$-plane.

Thus, the traces in the three coordinate planes are straight lines. The surface itself is a plane.

---

**(b)**  
We consider the surface  
$$
\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1.
$$

• In the $$xy$$-plane set $$z = 0$$. The equation reduces to  
$$
\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1.
$$
This is an ellipse in the $$xy$$-plane.

• In the $$xz$$-plane set $$y = 0$$. The equation becomes  
$$
\frac{x^{2}}{a^{2}} - \frac{z^{2}}{c^{2}} = 1.
$$
This is the equation of a hyperbola in the $$xz$$-plane.

• In the $$yz$$-plane set $$x = 0$$. The equation becomes  
$$
\frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1.
$$
This is also the equation of a hyperbola in the $$yz$$-plane.

The overall surface is known as a hyperboloid of one sheet. Its cross section in the horizontal ($$xy$$-) plane is an ellipse, while vertical cross sections (parallel to the $$xz$$- and $$yz$$-planes) are hyperbolas.

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**Sketching in 3D:**  
One would draw a plane in part (a) intersecting the coordinate planes along the straight lines we obtained. For part (b), sketch an elongated surface such that its horizontal slice is an ellipse (centered at the origin) and its vertical slices are hyperbolas. The hyperboloid of one sheet continuously “narrows” as one moves away from $$z=0$$ and then “flaring out” again, giving the characteristic one-sheet shape.
**(a)** The traces in the coordinate planes for the plane $$2x + 4y + 3z = 12$$ are straight lines. The surface is a plane. **(b)** For the surface $$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1$$, the traces in the coordinate planes are an ellipse in the $$xy$$-plane and hyperbolas in the $$xz$$- and $$yz$$-planes. The surface is a hyperboloid of one sheet. **Sketch:** - In part (a), draw a flat plane intersecting the coordinate planes along straight lines. - In part (b), sketch a hyperboloid of one sheet with an elliptical base and hyperbolic cross-sections along the vertical axes.

The traces in the coordinate planes refer to the curves you get when a surface
intersects one of the three coordinate planes in . What are the traces in the
coordinate planes for the following surfaces? Sketch them on a single space. Can
you name the surfaces?
(a)
(b)

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The traces in the coordinate planes refer to the curves you get when a surface intersects one of the three coordinate planes in . What are the traces in the coordinate planes for the following surfaces? Sketch them on a single space. Can you name the surfaces? (a) (b)