Question

Here's the question, please help!

Answer 1

• x=2sint
• y=3sint

#1

• x²/2+y²/3
• (2sint)²/2+(3sint)²/3
• 4sin²t/2+9sim²t/3
• 2sin²t+3sim²t
• 5sin²t

False

#2

• x²+y²
• 4sin²t+9sin²t
• 13sin²t

False

#3

• 3x²+2y²
• 3(4sin²t)+2(9sin²t)
• 12sin²+18sin²t
• 30sin²t

False

None of the above

Answer 2

Answer:

None of these

Step-by-step explanation:

To convert the parametric curve into Cartesian form,
rewrite the equation for $x$ to make $\sin t$ the subject:

$x=2\sin t$

$\implies \sin t=\dfrac{x}{2}$

Substitute this into the given equation for $y$:

$\begin{aligned}y & =3 \sin t\\\implies y & = 3 \left(\dfrac{x}{2}\right)\\y& = \dfrac{3}{2}x\end{aligned}$

Therefore, the Cartesian form of the parametric curve is:

$y=\dfrac{3}{2}x$

Further Information

$\dfrac{x^2}{2}+\dfrac{y^2}{3}=1 \quad \textsf{is the equation of a vertical ellipse}$

$\textsf{with center (0, 0), co-vertex }\sf \sqrt{2}, \textsf{ and vertex }\sqrt{3}$

$x^2+y^2=6 \quad \textsf{is the equation of a circle}$

$\textsf{with center (0, 0) and radius }\sf \sqrt{6}$

$3x^2+2y^2=1 \quad \textsf{is the equation of a vertical ellipse}$

$\textsf{with center (0, 0), co-vertex }\sf \dfrac{\sqrt{3}}{3}, \textsf{ and vertex }\dfrac{\sqrt{2}}{2}$