How do you convert $y=2y^2-x^2+6xy$ into a polar equation?

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Answer 1 · 2018-03-03T13:37:58

$r=sintheta/(2-3cos^2theta+3sin2theta)$

The relation between Cartesian coordinates $(x,y)$ and polar coordinates $(r,theta)$ is given by

$x=rcostheta$ , $y=rsintheta$ and $x^2+y^2=r^2$

Hence we can write $y=2y^2-x^2+6xy$ as

$rsintheta=2r^2sin^2theta-r^2cos^2theta+6r^2sinthetacostheta$

or $sintheta=r(2sin^2theta-cos^2theta+6sinthetacostheta)$

or $r=sintheta/(2sin^2theta-cos^2theta+6sinthetacostheta)$

= $sintheta/(2-3cos^2theta+3sin2theta)$

How do you convert y=2y^2-x^2+6xy y=2y^2-x^2+6xy into a polar equation?