Find an equation in rectangular coordinates for the spherical equation \( \rho=9 \cos (\phi) \)

To convert the spherical equation \( \rho = 9 \cos(\phi) \) into rectangular coordinates, we can use the relationships between spherical and rectangular coordinates: 1. \( x = \rho \sin(\phi) \cos(\theta) \) 2. \( y = \rho \sin(\phi) \sin(\theta) \) 3. \( z = \rho \cos(\phi) \) 4. \( \rho = \sqrt{x^2 + y^2 + z^2} \) Given the equation \( \rho = 9 \cos(\phi) \), we can express \( \cos(\phi) \) in terms of \( z \) and \( \rho \): \[ \cos(\phi) = \frac{z}{\rho} \] Substituting this into the original equation gives: \[ \rho = 9 \left(\frac{z}{\rho}\right) \] Multiplying both sides by \( \rho \) to eliminate the fraction: \[ \rho^2 = 9z \] Now, substituting \( \rho^2 \) with \( x^2 + y^2 + z^2 \): \[ x^2 + y^2 + z^2 = 9z \] Rearranging this equation gives: \[ x^2 + y^2 + z^2 - 9z = 0 \] To express this in a more standard form, we can complete the square for the \( z \) terms: \[ x^2 + y^2 + (z^2 - 9z) = 0 \] Completing the square for \( z^2 - 9z \): \[ z^2 - 9z = (z - \frac{9}{2})^2 - \frac{81}{4} \] Substituting this back into the equation: \[ x^2 + y^2 + \left(z - \frac{9}{2}\right)^2 - \frac{81}{4} = 0 \] Adding \( \frac{81}{4} \) to both sides gives: \[ x^2 + y^2 + \left(z - \frac{9}{2}\right)^2 = \frac{81}{4} \] This is the equation of a sphere with center at \( (0, 0, \frac{9}{2}) \) and radius \( \frac{9}{2} \). Thus, the equation in rectangular coordinates is: \[ x^2 + y^2 + \left(z - \frac{9}{2}\right)^2 = \frac{81}{4} \]