Answer
${\left( {x + y} \right)^2} = xy + 6$ as an equation in polar coordinates:
${r^2} = \frac{6}{{1 + \cos \theta \sin \theta }}$
Work Step by Step
Write
${\left( {x + y} \right)^2} = xy + 6$
${x^2} + 2xy + {y^2} = xy + 6$
${x^2} + {y^2} + xy = 6$
Since $x = r\cos \theta $ and $y = r\sin \theta $, we get ${x^2} + {y^2} = {r^2}$. So,
${r^2} + {r^2}\cos \theta \sin \theta = 6$
${r^2} = \frac{6}{{1 + \cos \theta \sin \theta }}$
