The parametric equation
`x=a(sec theta+tan theta),y=b(sec theta-tan theta)` repersents

Options

A. a parabola
B. an ellipse
C. a hyperbola
D. a rectangular hyperbola

Answer

To solve the problem, we need to analyze the given parametric equations:

1. **Given Parametric Equations**:
\[ x = a(\sec \theta + \tan \theta) \]
\[ y = b(\sec \theta - \tan \theta) \]

2. **Using Trigonometric Identity**:
We know the trigonometric identity:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
This can be rewritten as:
\[ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1 \]

3. **Substituting the Parametric Equations**:
From the parametric equations, we can express \(\sec \theta + \tan \theta\) and \(\sec \theta - \tan \theta\) in terms of \(x\) and \(y\):
\[ \sec \theta + \tan \theta = \frac{x}{a} \]
\[ \sec \theta - \tan \theta = \frac{y}{b} \]

4. **Multiplying the Two Expressions**:
Now, substituting these into the identity:
\[ \left(\frac{x}{a}\right)\left(\frac{y}{b}\right) = 1 \]

5. **Rearranging the Equation**:
This simplifies to:
\[ xy = ab \]

6. **Identifying the Type of Conic**:
The equation \(xy = ab\) represents a rectangular hyperbola, as it can be expressed in the form \(xy = c\), where \(c\) is a constant.

7. **Conclusion**:
Therefore, the parametric equations represent a rectangular hyperbola.

### Final Answer:
The parametric equations \(x = a(\sec \theta + \tan \theta)\) and \(y = b(\sec \theta - \tan \theta)\) represent a rectangular hyperbola.

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