Question

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

\({\rm{4}}{{\rm{y}}^{\rm{2}}}{\rm{ = x}}\)

Short answer

The substitution equation

\({\rm{r = }}\frac{{{\rm{cos\theta }}}}{{{\rm{4si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}}}\)

Step-by-step solution

Formulae that translate.

Formulas’ expressing a point’s Cartesian coordinates \(\left( {{\rm{x,y}}} \right)\)in terms of polar coordinates\(\left( {{\rm{r,\theta }}} \right){\rm{.}}\)

\(\begin{aligned}{l}{\rm{x = rcos\theta }}\\{\rm{y = rsin\theta }}\end{aligned}\)

 Replace that in the equation.        

\(\begin{aligned}{c}{\rm{4}}{{\rm{y}}^{\rm{2}}}{\rm{ = x4 \times (rsin\theta }}{{\rm{)}}^{\rm{2}}}{\rm{ }}\\{\rm{ = rcos\theta 4}}{{\rm{r}}^{\rm{2}}}{\rm{si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}\\{\rm{ = rcos\theta 4rsi}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}\\{\rm{ = cos\theta r }}\\{\rm{ = }}\frac{{{\rm{cos\theta }}}}{{{\rm{4si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}}}\end{aligned}\)

In the equation, substitute that\({\rm{r = }}\frac{{{\rm{cos\theta }}}}{{{\rm{4si}}{{\rm{n}}^{\rm{2}}}{\rm{\theta }}}}\)