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Section 9.6 : Polar Coordinates
6. Convert the following equation into an equation in terms of polar coordinates.
\[{x^2} = \frac{{4x}}{y} - 3{y^2} + 2\] Show SolutionBasically, what we need to do here is to convert all the \(x\)’s and \(y\)’s into \(r\)’s and \(\theta \)’s using the following formulas.
\[x = r\cos \theta \hspace{0.25in}\hspace{0.25in}y = r\sin \theta \hspace{0.25in}\hspace{0.25in}{r^2} = {x^2} + {y^2}\]Don’t forget about the last one! If it is possible to use this formula (which won’t do us a lot of good in this problem) it will save a lot of work!
First let’s substitute in the equations as needed.
\[{\left( {r\cos \theta } \right)^2} = \frac{{4\left( {r\cos \theta } \right)}}{{r\sin \theta }} - 3{\left( {r\sin \theta } \right)^2} + 2\]Finally, as we need to do is take care of little simplification to get,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{{r^2}{{\cos }^2}\theta = 4\cot \theta - 3{r^2}{{\sin }^2}\theta + 2}}\]