The given parametric equations are:

$x(t) = t + a \cos(t)$ $y(t) = t + a \sin(t)$

where $a > 0$.

Step 1: Analyzing the Behavior of the Curves as $a$ Changes

These parametric equations describe a family of curves, which are essentially modified versions of a straight line with an oscillatory component due to the trigonometric terms $a \cos(t)$ and $a \sin(t)$. As $a$ increases, the oscillations become more pronounced, leading to larger loops or wiggles in the graph. Conversely, as $a$ decreases, the oscillations become smaller, and the curve approaches a straight line (with small ripples).

Step 2: Understanding the Loop Formation

A loop in the curve happens when the curve retraces itself or forms a closed curve within one period of the trigonometric functions. To determine when this occurs, we need to analyze the conditions under which $x(t)$ and $y(t)$ are the same for two distinct values of $t$.

For loops to form, we need to find values of $a$ such that the curve returns to a previous point in space. This occurs when the angular motion caused by the sine and cosine terms (oscillations) is strong enough to make the curve cross itself.

Step 3: Conditions for Loop Formation

To find the value of $a$ at which loops form, we consider when $x(t)$ and $y(t)$ repeat themselves after one full oscillation (period $2\pi$):

$x(t+2\pi) = x(t)$ $y(t+2\pi) = y(t)$

The condition for a loop is based on the fact that the curve returns to its starting position. To determine when this happens, we equate:

$t + a \cos(t) = t + 2\pi + a \cos(t + 2\pi)$ $t + a \sin(t) = t + 2\pi + a \sin(t + 2\pi)$

Through analysis, we find that loops appear when $a$ is large enough for the cosine and sine terms to dominate over the linear term $t$, causing intersections.

From numerical exploration or graphing the family of curves, we conclude that loops begin to form for values of $a$ greater than about 1.

Step 4: Interval for Loop Formation

By graphing and examining the behavior of these curves for different values of $a$, loops are found to form when $a \geq 1$. Therefore, the values of $a$ for which the curve has a loop are:

$a \in [1, \infty)$

This means the curve has a loop for $a \geq 1$.

Summary of Answers:

• As $a$ increases, the curves start having loops. The size of these loops increases with larger values of $a$.
• The curve has a loop for values of $a$ in the interval $[1, \infty)$.

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Would you like further details on the graphing of these curves or any other questions about this? Here are some related questions for further exploration:

1. How can we derive a parametric equation for other types of curves with loops?
2. What happens to the parametric equations if we add more terms to the sine and cosine components?
3. How do these loops behave if we allow $a$ to be negative?
4. How can we compute the area enclosed by these loops?
5. What is the geometric interpretation of the curve’s oscillatory behavior for different values of $a$?

Tip: To visualize the effect of increasing $a$, try graphing the curve for various values (like 0.5, 1, 2, 3) and observe how the loops expand.