Write the equation 3𝑥2+4𝑧2=9 in spherical coordinates.

To convert the equation \(3x^2 + 4z^2 = 9\) into spherical coordinates, we'll use the following transformations for spherical coordinates \((r, \theta, \phi)\):

- \(x = r \sin\theta \cos\phi\)
- \(y = r \sin\theta \sin\phi\)
- \(z = r \cos\theta\)

Now let's substitute these into the equation \(3x^2 + 4z^2 = 9\):

1. Substitute \(x = r \sin\theta \cos\phi\) and \(z = r \cos\theta\):
   \[
   3(r \sin\theta \cos\phi)^2 + 4(r \cos\theta)^2 = 9
   \]

2. Simplify each term:
   \[
   3r^2 \sin^2\theta \cos^2\phi + 4r^2 \cos^2\theta = 9
   \]

3. Factor out \(r^2\):
   \[
   r^2(3 \sin^2\theta \cos^2\phi + 4 \cos^2\theta) = 9
   \]

4. Solve for \(r^2\):
   \[
   r^2 = \frac{9}{3 \sin^2\theta \cos^2\phi + 4 \cos^2\theta}
   \]

So, the equation \(3x^2 + 4z^2 = 9\) in spherical coordinates is:

\[
r^2 = \frac{9}{3 \sin^2\theta \cos^2\phi + 4 \cos^2\theta}
\]

Would you like more details on any step? Here are 5 related questions:

1. How do you derive spherical coordinates from Cartesian coordinates?
2. What is the physical interpretation of spherical coordinates?
3. Can this equation represent any specific surface in 3D space?
4. How would you convert the same equation to cylindrical coordinates?
5. How does the equation change if you have terms involving \(y^2\) as well?

**Tip:** Always check which coordinate system simplifies your equation or makes interpretation easier based on symmetry.

Write the equation 3x^2 + 4z^2 = 9 in spherical coordinates.

Explore the conversion of the equation 3x^2 + 4z^2 = 9 into spherical coordinates with detailed explanations. Learn about the coordinate transformations and the step-by-step process of substituting Cartesian variables into spherical coordinates. Suitable for advanced high school or college-level students.

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