Answer:
None of these
Step-by-step explanation:
To convert the parametric curve into Cartesian form,
rewrite the equation for [tex]x[/tex] to make [tex]\sin t[/tex] the subject:
[tex]x=2\sin t[/tex]
[tex]\implies \sin t=\dfrac{x}{2}[/tex]
Substitute this into the given equation for [tex]y[/tex]:
[tex]\begin{aligned}y & =3 \sin t\\\implies y & = 3 \left(\dfrac{x}{2}\right)\\y& = \dfrac{3}{2}x\end{aligned}[/tex]
Therefore, the Cartesian form of the parametric curve is:
[tex]y=\dfrac{3}{2}x[/tex]
Further Information
[tex]\dfrac{x^2}{2}+\dfrac{y^2}{3}=1 \quad \textsf{is the equation of a vertical ellipse}[/tex]
[tex]\textsf{with center (0, 0), co-vertex }\sf \sqrt{2}, \textsf{ and vertex }\sqrt{3}[/tex]
[tex]x^2+y^2=6 \quad \textsf{is the equation of a circle}[/tex]
[tex]\textsf{with center (0, 0) and radius }\sf \sqrt{6}[/tex]
[tex]3x^2+2y^2=1 \quad \textsf{is the equation of a vertical ellipse}[/tex]
[tex]\textsf{with center (0, 0), co-vertex }\sf \dfrac{\sqrt{3}}{3}, \textsf{ and vertex }\dfrac{\sqrt{2}}{2}[/tex]
