IF t is a parameter, then `x = a(t + (1)/(t))` and `y = b(t - (1)/(t))` represents

Options

A. An ellipse
B. A parabola
C. A hyperbola
D. A circle

Answer

To solve the problem, we start with the given parametric equations:

1. \( x = a\left(t + \frac{1}{t}\right) \)
2. \( y = b\left(t - \frac{1}{t}\right) \)

### Step 1: Rearranging the Equations
We can rearrange both equations to isolate \( t \):

- From the equation for \( x \):
\[ \frac{x}{a} = t + \frac{1}{t} \]

- From the equation for \( y \):
\[ \frac{y}{b} = t - \frac{1}{t} \]

### Step 2: Adding and Subtracting the Equations
Now, we will add and subtract these two equations:

- **Adding**:
\[ \frac{x}{a} + \frac{y}{b} = \left(t + \frac{1}{t}\right) + \left(t - \frac{1}{t}\right) = 2t \]
This gives us:
\[ \frac{x}{a} + \frac{y}{b} = 2t \quad \text{(1)} \]

- **Subtracting**:
\[ \frac{x}{a} - \frac{y}{b} = \left(t + \frac{1}{t}\right) - \left(t - \frac{1}{t}\right) = 2\frac{1}{t} \]
This gives us:
\[ \frac{x}{a} - \frac{y}{b} = \frac{2}{t} \quad \text{(2)} \]

### Step 3: Expressing \( t \)
From equation (1), we can express \( t \):
\[ t = \frac{1}{2}\left(\frac{x}{a} + \frac{y}{b}\right) \]

### Step 4: Substituting \( t \) into Equation (2)
Now, we substitute \( t \) back into equation (2):
\[ \frac{x}{a} - \frac{y}{b} = \frac{2}{\frac{1}{2}\left(\frac{x}{a} + \frac{y}{b}\right)} \]
This simplifies to:
\[ \frac{x}{a} - \frac{y}{b} = \frac{4}{\left(\frac{x}{a} + \frac{y}{b}\right)} \]

### Step 5: Cross-Multiplying
Cross-multiplying gives:
\[ \left(\frac{x}{a} - \frac{y}{b}\right)\left(\frac{x}{a} + \frac{y}{b}\right) = 4 \]
This can be rewritten using the difference of squares:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 4 \]

### Step 6: Final Form
To express this in standard form, we divide through by 4:
\[ \frac{x^2}{4a^2} - \frac{y^2}{4b^2} = 1 \]

This is the standard form of the equation of a hyperbola.

### Conclusion
Thus, the given parametric equations represent a hyperbola.