How do you eliminate the parameter to find a cartesian equation of the curve?

Question

$$x=1/2cosθ$$ $$y=2sinθ$$ $$0 \le θ \le π $$

So I know the parameter that must be eliminated is θ. How should I do this? Are there trig identities that I can use?

Answer 1

Yes, you can use $\cos^2\theta+\sin^2\theta=1$.

First, represent $\cos\theta,\sin\theta$ by $x,y$ respectively. Then, use $\cos^2\theta+\sin^2\theta=1$ to eliminate $\theta$.

Answer 2

$2x = \cos \theta$ and $y=\sin \theta$ so $(2x)^2 + y^2 =1$ or $4 x^2 + y^2 = 1$. Note the domain $0 \le \theta \le \pi$ means $\sin \theta \ge 0$, that is $y \ge 0$. Therefore:

\begin{eqnarray*} 4 x^2 + y^2 = 1\ \text{and } y \ge 0 \end{eqnarray*}