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

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Answer 1 · 2017-06-06T06:52:18

The polar equation is #r=(costheta-2sintheta)/(sinthetacostheta+3sin^2theta)#

To convert from rectangular coordinates #(x,y)# to polar coordinates #(r,theta)#, we apply the following equations

#x=rcostheta#

#y=rsintheta#

Therefore,

#xy=x-2y-3y^2#

#rcostheta*rsintheta=rcostheta-2rsintheta-3r^2sin^2theta#

#r^2costhetasintheta=rcostheta-2rsintheta-3r^2sin^2theta#

As #r!=0#, we divide by #r#

#rcosthetasintheta=costheta-2sintheta-3rsin^2theta#

#rsinthetacostheta+3rsin^2theta=costheta-2sintheta#

#r(sinthetacostheta+3sin^2theta)=costheta-2sintheta#

So,

#r=(costheta-2sintheta)/(sinthetacostheta+3sin^2theta)#