Let's break down the given parametric equations and analyze the statements provided:
1. The parametric equations are:
[tex]\[
\begin{array}{l}
x = 2 \cos(t) \\
y = \sin(t)
\end{array}
\][/tex]
Here, [tex]\( t \)[/tex] is the parameter that describes the curve.
2. Let's examine each statement given:
- Statement 1: "The parameter is [tex]\( x \)[/tex] and the curve contains the points [tex]\( y = \sin(t) \)[/tex]."
- Here, the parameter is stated to be [tex]\( x \)[/tex]. However, in our parametric equations, [tex]\( x \)[/tex] is not the parameter; [tex]\( t \)[/tex] is. Hence, this statement is incorrect.
- Statement 2: "The parameter is [tex]\( y \)[/tex] and the curve contains the points [tex]\( x = 2 \cos(t) \)[/tex]."
- In this statement, [tex]\( y \)[/tex] is claimed to be the parameter. But again, in our parametric equations, [tex]\( y \)[/tex] is not the parameter; [tex]\( t \)[/tex] is. Thus, this statement is also incorrect.
- Statement 3: "The parameter is [tex]\( t \)[/tex] and the curve contains the set of points [tex]\( (2 \cos(t), \sin(t)) \)[/tex]."
- Here, the parameter is correctly identified as [tex]\( t \)[/tex]. The set of points described [tex]\( (2 \cos(t), \sin(t)) \)[/tex] matches the parametric equations given. Thus, this statement is correct.
- Statement 4: "The parameter is [tex]\( t \)[/tex] and the curve contains the points [tex]\( (\sin(t), 2 \cos(t)) \)[/tex]."
- Although [tex]\( t \)[/tex] is correctly identified as the parameter, the points described [tex]\( (\sin(t), 2 \cos(t)) \)[/tex] do not match the order given by the parametric equations. The correct order is [tex]\( (2 \cos(t), \sin(t)) \)[/tex]. Hence, this statement is incorrect.
Given this analysis, the statement that best describes the curve is:
The parameter is [tex]\( t \)[/tex] and the curve contains the set of points [tex]\( (2 \cos(t), \sin(t)) \)[/tex].
Therefore, the correct choice is:
Option 3: The parameter is [tex]\( t \)[/tex] and the curve contains the set of points [tex]\( (2 \cos(t), \sin(t)) \)[/tex].
