 Find the equations of the​ xy-, xz-, and​ yz-traces, if they exist. of this equation x^2 + (y^2/256) + (z^2/256) = 1

The given equation is:

\[
x^2 + \frac{y^2}{256} + \frac{z^2}{256} = 1
\]

We can find the traces of this surface in the **xy-, xz-, and yz-planes** by setting one of the variables to zero at a time.

### 1. **xy-trace** (set \( z = 0 \))
To find the trace in the **xy-plane**, we set \( z = 0 \) in the equation:

\[
x^2 + \frac{y^2}{256} = 1
\]

This is the equation of an **ellipse** in the **xy-plane**, where the semi-major axis is along the \( y \)-axis with a length of 16 (since \( \sqrt{256} = 16 \)), and the semi-minor axis is along the \( x \)-axis with a length of 1 (since \( \sqrt{1} = 1 \)).

### 2. **xz-trace** (set \( y = 0 \))
To find the trace in the **xz-plane**, we set \( y = 0 \) in the equation:

\[
x^2 + \frac{z^2}{256} = 1
\]

This is also the equation of an **ellipse** in the **xz-plane**, with the same properties as the ellipse in the **xy-plane**. The semi-major axis is along the \( z \)-axis with a length of 16, and the semi-minor axis is along the \( x \)-axis with a length of 1.

### 3. **yz-trace** (set \( x = 0 \))
To find the trace in the **yz-plane**, we set \( x = 0 \) in the equation:

\[
\frac{y^2}{256} + \frac{z^2}{256} = 1
\]

This is the equation of a **circle** in the **yz-plane** with a radius of 16, as both the \( y \)-axis and the \( z \)-axis have the same coefficient of 256.

### Summary of traces:
1. **xy-trace**: \( x^2 + \frac{y^2}{256} = 1 \) (ellipse)
2. **xz-trace**: \( x^2 + \frac{z^2}{256} = 1 \) (ellipse)
3. **yz-trace**: \( \frac{y^2}{256} + \frac{z^2}{256} = 1 \) (circle)

Would you like any details or have any further questions?

Here are some related questions:
1. How do you find traces for surfaces of different types of quadratic equations?
2. What is the difference between an ellipse and a hyperbola when looking at traces?
3. How would the traces change if the equation was \( x^2 + \frac{y^2}{a} + \frac{z^2}{b} = 1 \)?
4. What does the geometric interpretation of traces tell you about the shape of the surface?
5. How can traces help in sketching three-dimensional surfaces?

**Tip:** When analyzing conic sections (like ellipses, circles, and hyperbolas), always check the coefficients of \( x \), \( y \), and \( z \) in the equation to identify the axis lengths.

Find the equations of the xy-, xz-, and yz-traces, if they exist, of this equation x^2 + (y^2/256) + (z^2/256) = 1

Explore how to find the equations of the xy-, xz-, and yz-traces of the surface defined by x^2 + (y^2/256) + (z^2/256) = 1. This problem involves concepts of ellipses and circles, making it suitable for advanced high school students studying conic sections.

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