Question

In Exercises \(9-14,\) find an equation in cylindrical coordinates for the equation given in rectangular coordinates. $$ x^{2}+y^{2}=8 x $$

Short answer

The equation in cylindrical coordinates is \(r = 8\cos(\theta)\).

Step-by-step solution

Identify the given rectangular coordinates equation

The equation given in rectangular coordinates is \(x^{2}+y^{2}=8 x\)

Write down the transformation equations

For cylindrical coordinates, the transformation equations from rectangular coordinates are as follows: \(x = r \cos(\theta)\), \(y = r \sin(\theta)\), \(z = z\)

Substitute into the given equation

Substituting these into the equation \(x^{2}+y^{2}=8 x\), we get \((r \cos(\theta))^{2} + (r \sin(\theta))^{2} = 8r\cos(\theta)\), which simplifies to \(r^{2} = 8r \cos(\theta)\)

Simplify the equation

This further simplifies to \(r = 8 \cos(\theta)\), which is the equation of the given function in cylindrical coordinates