Find implicit equation of the following plane parametrized curves

Question

I have the following parametrized plane curve: $C_1 =$ Image of $c_1$, where $c_1(t)=(2\cos t,3\sin t)$, and I need to find implicit equation. I don't know exactly what are implicit equation. This exercise is solved in my book and the answare is $C_1:\frac{x(t)^2}{4} + \frac{y(t)^2}{9} = 1$. I don't know why is this the answare.

Answer 1

Let use write the curve in component: $x$ and $y$

Thus:

$\begin{align} x(t)=2\cos(t)\\ y(t)=3\sin(t)\\ \frac{x(t)}{2}=\cos(t)\\ \frac{y(t)}{3}=\sin(t)\\ \left(\frac{x(t)}{2}\right)^2+\left(\frac{y(t)}{3}\right)^2=\\ \sin^2(t)+\cos^2(t)=\\ 1 \end{align}$

Following the equations, we have just shown that your curve $C$ is described by:

$\begin{equation} \frac{x(t)^2}{4}+\frac{y(t)^2}{9}=1 \end{equation}$