We have three problems: 1. Convert $x^2 - y^2 = 25$ to cylindrical coordinates.

1. Problem Description

We have three problems:

1. Convert $x^2 - y^2 = 25$ to cylindrical coordinates.

2. Convert $2x^2 + 2y^2 - 4z^2 = 0$ to spherical coordinates.

3. Convert $x^2 + y^2 = 9$ to spherical coordinates.

2. Solution Steps

Problem 1: $x^2 - y^2 = 25$ to cylindrical coordinates.

In cylindrical coordinates, we have the following relationships:

$x = r \cos(\theta)$

$y = r \sin(\theta)$

$z = z$

Substituting these into the equation, we get:

$(r \cos(\theta))^2 - (r \sin(\theta))^2 = 25$

$r^2 \cos^2(\theta) - r^2 \sin^2(\theta) = 25$

$r^2 (\cos^2(\theta) - \sin^2(\theta)) = 25$

$r^2 \cos(2\theta) = 25$

Problem 2: $2x^2 + 2y^2 - 4z^2 = 0$ to spherical coordinates.

In spherical coordinates, we have the following relationships:

$x = \rho \sin(\phi) \cos(\theta)$

$y = \rho \sin(\phi) \sin(\theta)$

$z = \rho \cos(\phi)$

Substituting these into the equation, we get:

$2(\rho \sin(\phi) \cos(\theta))^2 + 2(\rho \sin(\phi) \sin(\theta))^2 - 4(\rho \cos(\phi))^2 = 0$

$2\rho^2 \sin^2(\phi) \cos^2(\theta) + 2\rho^2 \sin^2(\phi) \sin^2(\theta) - 4\rho^2 \cos^2(\phi) = 0$

$2\rho^2 \sin^2(\phi) (\cos^2(\theta) + \sin^2(\theta)) - 4\rho^2 \cos^2(\phi) = 0$

$2\rho^2 \sin^2(\phi) (1) - 4\rho^2 \cos^2(\phi) = 0$

$2\rho^2 \sin^2(\phi) = 4\rho^2 \cos^2(\phi)$

$2 \sin^2(\phi) = 4 \cos^2(\phi)$

$\sin^2(\phi) = 2 \cos^2(\phi)$

$\frac{\sin^2(\phi)}{\cos^2(\phi)} = 2$

$\tan^2(\phi) = 2$

$\tan(\phi) = \sqrt{2}$

$\phi = \arctan(\sqrt{2})$

Problem 3: $x^2 + y^2 = 9$ to spherical coordinates.

In spherical coordinates, we have:

$x = \rho \sin(\phi) \cos(\theta)$

$y = \rho \sin(\phi) \sin(\theta)$

$z = \rho \cos(\phi)$

Substituting into the equation, we get:

$(\rho \sin(\phi) \cos(\theta))^2 + (\rho \sin(\phi) \sin(\theta))^2 = 9$

$\rho^2 \sin^2(\phi) \cos^2(\theta) + \rho^2 \sin^2(\phi) \sin^2(\theta) = 9$

$\rho^2 \sin^2(\phi) (\cos^2(\theta) + \sin^2(\theta)) = 9$

$\rho^2 \sin^2(\phi) (1) = 9$

$\rho^2 \sin^2(\phi) = 9$

3. Final Answer

1. $r^2 \cos(2\theta) = 25$

2. $\phi = \arctan(\sqrt{2})$

3. $\rho^2 \sin^2(\phi) = 9$