2. Given $$ \frac{x^{2}}{9}+y^{2}+\frac{z^{2}}{9}=1 $$. Use trace and level technique to sketch the equation in three dimension.

Question

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The equation given is

$$
\frac{x^{2}}{9} + y^{2} + \frac{z^{2}}{9} = 1.
$$

This equation represents an ellipsoid centered at the origin in three-dimensional space. To understand its shape and sketch it, we can analyze the equation using the trace and level technique.

### Step 1: Identify the Axes

The equation can be rewritten in a more standard form:

$$
\frac{x^{2}}{3^2} + \frac{y^{2}}{1^2} + \frac{z^{2}}{3^2} = 1.
$$

From this, we can identify the semi-axis lengths:

- The semi-axis along the $$x$$ direction is $$3$$ (since $$3^2 = 9$$).
- The semi-axis along the $$y$$ direction is $$1$$ (since $$1^2 = 1$$).
- The semi-axis along the $$z$$ direction is $$3$$ (since $$3^2 = 9$$).

### Step 2: Traces in the Coordinate Planes

To visualize the ellipsoid, we can take traces in the coordinate planes by setting one variable to a constant value.

1. **Trace in the $$xy$$-plane** ($$z = 0$$):
   $$
   \frac{x^{2}}{9} + y^{2} = 1.
   $$
   This is an ellipse with semi-major axis $$3$$ along the $$x$$-axis and semi-minor axis $$1$$ along the $$y$$-axis.

2. **Trace in the $$xz$$-plane** ($$y = 0$$):
   $$
   \frac{x^{2}}{9} + \frac{z^{2}}{9} = 1.
   $$
   This is also an ellipse with semi-major axis $$3$$ along the $$x$$-axis and semi-major axis $$3$$ along the $$z$$-axis.

3. **Trace in the $$yz$$-plane** ($$x = 0$$):
   $$
   y^{2} + \frac{z^{2}}{9} = 1.
   $$
   This is an ellipse with semi-minor axis $$1$$ along the $$y$$-axis and semi-major axis $$3$$ along the $$z$$-axis.

### Step 3: Sketching the Ellipsoid

Now that we have the traces, we can sketch the ellipsoid:

- The ellipsoid is elongated along the $$x$$ and $$z$$ axes (both have a length of $$3$$), and it is shorter along the $$y$$ axis (length of $$1$$).
- The shape will look like a stretched sphere, wider in the $$x$$ and $$z$$ directions and narrower in the $$y$$ direction.

### Summary of the Sketch

1. Draw the three axes: $$x$$, $$y$$, and $$z$$.
2. Mark the lengths of the semi-axes:
   - $$x$$ from $$-3$$ to $$3$$,
   - $$y$$ from $$-1$$ to $$1$$,
   - $$z$$ from $$-3$$ to $$3$$.
3. Sketch the ellipsoid, ensuring it is wider along the $$x$$ and $$z$$ axes and narrower along the $$y$$ axis.

This will give you a good representation of the ellipsoid defined by the equation.
The equation represents an ellipsoid centered at the origin. The semi-axis lengths are 3 along the x and z axes, and 1 along the y axis. Sketch the ellipsoid by drawing the axes, marking the semi-axis lengths, and ensuring the ellipsoid is wider along the x and z axes and narrower along the y axis.

Given . Use trace and level technique to sketch the equation
in three dimension.

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Solución

Step 1: Identify the Axes

Step 2: Traces in the Coordinate Planes

Step 3: Sketching the Ellipsoid

Summary of the Sketch

Bonus Knowledge

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Given . Use trace and level technique to sketch the equation in three dimension.