Question

Find a polar equation for the curve represented by the given Cartesian equation.

\({{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ = 2cx}}\)

Short answer

Whatever the value of \({\rm{\theta }}\) is, \({\rm{r = 0}}\)represents the pole.

Step-by-step solution

Conversion of a Cartesian equation.

\({{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ = 2cx}}\)

Polar coordinates will be used in the polar equation.

\(\begin{aligned}{c}{\rm{x = rcos\theta ,}}\;\\\;{\rm{y = rsin\theta }}\end{aligned}\)

Formula\({{\rm{r}}^{\rm{2}}}{\rm{ = }}{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}\)

Regardless of whether or not \({\rm{\theta }}\) is a positive or negative number?

\(\begin{aligned}{c}{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ = 2cx}}\\{{\rm{(rcos\theta )}}^{\rm{2}}}{\rm{ + (rsin\theta }}{{\rm{)}}^{\rm{2}}}{\rm{ = 2crcos\theta }}\\{{\rm{r}}^{\rm{2}}}{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{\theta + }}{{\rm{r}}^{\rm{2}}}{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta = 2crcos\theta }}{{\rm{r}}^{\rm{2}}}\\{\rm{(cos\theta + sin\theta ) = 2crcos\theta }}\\{{\rm{r}}^{\rm{2}}}{\rm{ = 2crcos\theta }}\\{\rm{r = 2ccos\theta ,}}\;\;\\\;{\rm{r = 0}}\end{aligned}\)

\({\rm{r = 0}}\) \({\rm{r = 0}}\)Denotes the pole, regardless of its value\({\rm{\theta }}\).