Question

What the is the polar form of `y = y^2/x+(x-3)(y-5)`?

Answer 1

`r(-sinthetatantheta-rsinthetacostheta+4sintheta+5costheta)=15`

Explanation:

First, we expand everything to get:
`y=y^2/x+xy-3y-5y+15`

Now we need to use these:
`x=rcostheta`
`y=rsintheta`

`rsintheta=(r^2sin^2theta)/(rcostheta)+rcosthetarsintheta-3rsintheta-5rcostheta+15`

`rsintheta=rsinthetatantheta+r^2sinthetacostheta-3rsintheta-5rcostheta+15`

`rsintheta-rsinthetatantheta-r^2sinthetacostheta+3rsintheta+5rcostheta=15`

`r(-sinthetatantheta-rsinthetacostheta+4sintheta+5costheta)=15`

We cannot simplify this any further, so it stays as an implicit polar equation.