Question

Write the following parametric equations as a polar equation.x = 2ty=t²

Answer 1

ANSWER:

2nd option: r = 4 tan θ sec θ

STEP-BY-STEP EXPLANATION:

We have the following:

$\begin{gathered} x=2t\rightarrow t=\frac{x}{2} \\ \\ y=t^2 \end{gathered}$

We substitute the first equation in the second and we are left with the following:

$\begin{gathered} y=\left(\frac{x}{2}\right)^2 \\ \\ y=\frac{x^2}{2^2}=\frac{x^2}{4} \end{gathered}$

Now, we convert this to polar coordinates, just like this:

$\begin{gathered} x=r\cos\theta,y=r\sin\theta \\ \\ \text{ We replacing:} \\ \\ r\sin\theta=\frac{(r\cos\theta)^2}{4} \\ \\ r\sin\theta=\frac{r^2\cos^2\theta^{}}{4} \\ \\ r\sin\theta=\frac{r^2\cos\theta\cdot\cos\theta{}}{4} \\ \\ \frac{r^2\cos\theta\cdot\cos\theta}{4}=r\sin\theta \\ \\ r=4\frac{\sin\theta}{\cos\theta}\cdot\frac{1}{\cos\theta} \\ \\ r=4\tan\theta\cdot\sec\theta \end{gathered}$

So the correct answer is the 2nd option: r = 4 tan θ sec θ