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

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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:

$$
x^2 + y^2 + \left(z - \frac{9}{2}\right)^2 = \frac{81}{4}
$$
The equation in rectangular coordinates is $$ x^2 + y^2 + \left(z - \frac{9}{2}\right)^2 = \frac{81}{4} $$.

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

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