We want to rewrite the following parametric equations
[tex]\begin{gathered} \begin{cases}x={3\cos\theta} \\ y={6\sin\theta}\end{cases} \\ \frac{\pi}{2}<\theta<\frac{3\pi}{2} \end{gathered}[/tex]
as one equation. Using the following property
[tex]\sin^2\theta+\cos^2\theta=1[/tex]
We can eliminate the parameter theta adding the square of the coordinates
[tex]\begin{gathered} \sin^2\theta+\cos^2\theta=1 \\ 6^2(\sin^2\theta+\cos^2\theta)=6^2 \\ 6^2\sin^2\theta+6^2\cos^2\theta=6^2 \\ (6\sin\theta)^2+(6\cos\theta)^2=6^2 \\ (6\sin\theta)^2+(2\cdot3\cos\theta)^2=6^2 \\ (y)^2+(2x)^2=6^2 \\ y^2+4x^2=36 \\ \frac{x^2}{9}+\frac{y^2}{36}=1 \end{gathered}[/tex]
And this is the standard equation of an ellipse
[tex]\frac{x^{2}}{9}+\frac{y^{2}}{36}=1[/tex]
The constrain
[tex]\frac{\pi}{2}<\theta<\frac{3\pi}{2}\implies x<0[/tex]
tells us that x can only assume negative values, therefore, the graph is only the left side of the ellipse.
