91Ó°ÊÓ

Parametric equations are a set of equations that express the coordinates of points on a curve as functions of a variable, typically denoted by \( t \) (often called the parameter). In our original exercise, the parametric equations provided are:

• \( x = \frac{t}{t-1} \)
• \( y = \frac{1}{\sqrt{t-1}} \)

These equations give us the \( x \) and \( y \) coordinates in terms of the parameter \( t \), which ranges over a specified interval. Here, \( t \) is in the interval \((1, \infty)\), meaning we only consider values of \( t \) greater than 1.
This representation can be advantageous as it sometimes makes it easier to describe complex curves. The challenge is to eliminate the parameter \( t \) to obtain a direct relationship between \( x \) and \( y \), known as the rectangular or Cartesian equation.

Determining the interval for a variable like \( x \) in parametric equations involves considering the specified interval for the parameter \( t \). Our task is to understand how \( x \) behaves as \( t \) varies from 1 to infinity in the given equations:

• Equation: \( x = \frac{t}{t-1} \)

As \( t \to 1^+ \) (meaning \( t \) approaches 1 from the right), the expression \( t-1 \) approaches zero, causing \( x \) to rapidly increase towards infinity. On the other end, as \( t \to \infty \), the fraction \( \frac{t}{t-1} \) simplifies to \( \approx 1 \), as the subtraction becomes negligible for large \( t \). This indicates that \( x \) decreases towards 1.
Therefore, \( x \) spans the interval \((1, \infty)\), which means \( x \) only takes on values greater than 1, headed towards infinity. Understanding these intervals is crucial as it helps us map the parametric curve accurately onto its Cartesian form.

In the step-by-step approach to finding the rectangular equation, we aim to eliminate the parameter \( t \):1. **Express \( t \) in terms of \( x \):** Start with \( x = \frac{t}{t-1} \) and rearrange to solve for \( t \). This involves reorganizing the terms, factoring, and ultimately solving for \( t \) in terms of \( x \): \[ t = \frac{-x}{1-x} \] Knowing \( t \) allows us to substitute it into the equation for \( y \).2. **Substitute \( t \) into \( y \):** Use the expression found to replace \( t \) in \( y = \frac{1}{\sqrt{t-1}} \): \[ y = \sqrt{1-x} \] Through substitution and simplification, we arrive at a simpler relationship.3. **Combine for the rectangular equation:** With the substituted \( y = \sqrt{1-x} \), rewriting gives us the rectangular form: \[ y^2 = 1-x \] This new equation describes the curve in terms of \( x \) and \( y \) directly, without involving \( t \). This simplification process is crucial for graphing and analyzing the relationship between \( x \) and \( y \) without referencing the original parameter.