Question

Identify the curve by finding a Cartesian equation for the curve.

\({\rm{r = tan\theta sec\theta }}\)

Short answer

\({{\rm{x}}^{\rm{2}}}{\rm{ = y}}\)this region is parabola.

Step-by-step solution

Find a Cartesian equation for the curve and use it to identify it.

Rewrite with \({\rm{sin\theta }}\)and\({\rm{cos\theta }}\).

\(\begin{aligned}{c}{\rm{r = tan\theta sec\theta }}\\{\rm{ = }}\frac{{{\rm{sin\theta }}}}{{{\rm{cos\theta }}}}{\rm{*}}\frac{{\rm{1}}}{{{\rm{cos\theta }}}}\\{\rm{ = }}\frac{{{\rm{sin\theta }}}}{{{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{\theta }}}}\end{aligned}\)

Using\({\rm{co}}{{\rm{s}}^2}{\rm{\theta }}\), divide both sides.

\({\rm{rco}}{{\rm{s}}^{\rm{2}}}{\rm{\theta = sin\theta }}\)

Using\({\rm{r}}\), multiply both sides.

\({{\rm{r}}^{\rm{2}}}{\rm{co}}{{\rm{s}}^{\rm{2}}}{\rm{\theta = rsin\theta }}\)

Result.

Rewrite

\({\left( {{\rm{rcos\theta }}} \right)^{\rm{2}}}{\rm{ = rsin\theta }}\)

Substitute \({\rm{x = rcos\theta \& y = rsin\theta }}\)

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

Therefore, \({{\rm{x}}^{\rm{2}}}{\rm{ = y}}\)this region is parabola.