The graph of the curve represented by `{x=sectheta, y=costheta}` is

Options

A. a line
B. A hyperbola
C. an ellipse
D. a portion of hyperbola

Explanation

To solve the problem, we need to eliminate the parameter \(\theta\) from the equations \(x = \sec \theta\) and \(y = \cos \theta\) to find the relationship between \(x\) and \(y\).

### Step-by-Step Solution:

1. **Start with the given equations**:
\[ x = \sec \theta \quad \text{and} \quad y = \cos \theta \]

2. **Express \(\sec \theta\) in terms of \(\cos \theta\)**:
Recall that \(\sec \theta = \frac{1}{\cos \theta}\). Therefore, we can rewrite the equation for \(x\):
\[ x = \frac{1}{y} \]

3. **Rearranging the equation**:
Multiply both sides by \(y\) to eliminate the fraction:
\[ xy = 1 \]

4. **Identify the type of curve**:
The equation \(xy = 1\) represents a hyperbola.

5. **Determine the restrictions on \(y\)**:
Since \(y = \cos \theta\), the range of \(y\) is limited to \([-1, 1]\). Thus, \(y\) cannot take values outside this interval.

6. **Conclusion about the graph**:
The equation \(xy = 1\) describes a hyperbola, but due to the restriction on \(y\) (i.e., \(y\) can only be between -1 and 1), we are only looking at a portion of the hyperbola that lies within these bounds.

### Final Answer:
The graph of the curve represented by \(x = \sec \theta\) and \(y = \cos \theta\) is a **portion of a hyperbola**.

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