This is a parametric equation, where $t$ is the parameter. The equations provided are:

$x = \sqrt{1 - t^2}, \quad y = \frac{1}{t}.$

Goal:

The typical task is to eliminate the parameter $t$ to find a Cartesian equation, or analyze the curve formed by this parametric system.

Step 1: Square $x$

From the first equation: $x = \sqrt{1 - t^2} \implies x^2 = 1 - t^2.$ This implies: $t^2 = 1 - x^2. \quad \text{(1)}$

Step 2: Substitute $t^2$ into $y$

From the second equation: $y = \frac{1}{t} \implies t = \frac{1}{y}.$ Squaring both sides: $t^2 = \frac{1}{y^2}. \quad \text{(2)}$

Step 3: Combine Equations

Equating $t^2$ from $(1)$ and $(2)$: $1 - x^2 = \frac{1}{y^2}.$ Rearranging: $y^2(1 - x^2) = 1.$ Thus, the Cartesian equation is: $x^2 + y^2 x^2 = 1.$

Final Answer:

The Cartesian form of the curve is: $x^2 + x^2y^2 = 1.$

Would you like further clarification or visualizations for this?

Here are 5 related questions to explore:

1. What is the domain and range of $x = \sqrt{1 - t^2}$?
2. What happens to $y$ as $t \to 0$?
3. How do you parametrize this curve differently if $t > 0$?
4. Can this Cartesian equation be transformed into polar coordinates?
5. What type of curve does $x^2 + x^2y^2 = 1$ represent?

Tip: Always verify the domain of $t$ since the square root and division by zero impose restrictions.