Question

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

\({\rm{y = 1 + 3 x}}\)

Short answer

This is the polar equation.

\({\rm{r = }}\frac{{\rm{1}}}{{{\rm{sin\theta - 3cos\theta }}}}\)

Step-by-step solution

To figure out what the polar equation.

\({\rm{y = 1 + 3 x}}\)

Replace\({\rm{x}}\)with\({\rm{rcos\theta }}\)and\({\rm{y}}\)with\({\rm{rsin\theta }}\)the previous equation to find the polar equation.

\({\rm{rsin\theta = 1 + 3rcos\theta }}\)

Subtract\({\rm{3rcos\theta }}\)from both sides

\({\rm{rsin\theta - 3rcos\theta = 1}}\)

Remove \({\rm{r}}\)as a common element on the left side.

\({\rm{r(sin\theta - 3cos\theta ) = 1}}\)

\({\rm{\;sin\theta - 3cos\theta }}\)Divide both sides (this is allowed because it is non-zero, the above equation is not true when it happens to be\({\rm{0}}\))

\({\rm{r = }}\frac{{\rm{1}}}{{{\rm{sin\theta - 3cos\theta }}}}\)

This is the polar equation we're after\({\rm{r = }}\frac{{\rm{1}}}{{{\rm{sin\theta - 3cos\theta }}}}\).

\({\rm{r = }}\frac{{\rm{1}}}{{{\rm{sin\theta - 3cos\theta }}}}.\)

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