Question

Identify the curve by finding its Cartesian equation.
r = tan(theta) sec(theta) .

Answer 1

\[r=\tan\theta\sec\theta=\frac{\sin\theta}{\cos\theta}\sec\theta=\frac{\sin\theta}{\cos^2\theta}=\frac{\sin\theta}{1-\sin^2\theta}\]
Recall that \(y=r\sin\theta\), so
\[r=\frac{\frac{y}{r}}{1-\frac{y^2}{r^2}}\\
r=\frac{\frac{y}{r}}{1-\frac{y^2}{r^2}}\cdot\frac{r^2}{r^2}\\
r=\frac{ry}{r^2-y^2}\\
1=\frac{y}{r^2-y^2}\]
Also recall that \(r^2=x^2+y^2\):
\[1=\frac{y}{x^2+y^2-y^2}~~\Rightarrow~~1=\frac{y}{x^2}~~\Rightarrow~~y=x^2\]