Question

How do you convert `y=-y^2-3x^2-xy` into a polar equation?

Answer 1

`r=-(sintheta)/(sin^2theta+3cos^2theta+costhetasintheta)`

Explanation:

Rewrite as:
`y^2+3x^2+xy=-y`

Substitute in:
`x=rcostheta`
`y=rsintheta`

`(rsintheta)^2+3(rcostheta)^2+(rcostheta)(rsintheta)=-rsintheta`

`r^2(sintheta)^2+3r^2(costheta)^2+r^2(costhetasintheta)=-rsintheta`

Divide both sides by `r`

`r(sintheta)^2+3r(costheta)^2+r(costhetasintheta)=-sintheta`

Factorise out `r`:
`r(sin^2theta+3cos^2theta+costhetasintheta)=-sintheta`

Make `r` the subject:
`r=-(sintheta)/(sin^2theta+3cos^2theta+costhetasintheta)`