If `x=psectheta and y=qtantheta` then

Options

A. `x^2-y^2=p^2q^2`
B. `x^2q^2-y^2p^2=pq`
C. `x^2q^2-y^2p^2=1/(p^2q^2)`
D. `x^2q^2-y^2p^2=p^2q^2`

Answer

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

1. **Given Equations**:
- \( x = P \sec \theta \)
- \( y = Q \tan \theta \)

2. **Rearranging the Equations**:
- From the first equation, we can express \( \sec \theta \) in terms of \( x \) and \( P \):
\[ \sec \theta = \frac{x}{P} \]
- From the second equation, we can express \( \tan \theta \) in terms of \( y \) and \( Q \):
\[ \tan \theta = \frac{y}{Q} \]

3. **Using the Trigonometric Identity**:
- We know the identity:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
- Substituting the expressions for \( \sec \theta \) and \( \tan \theta \) into this identity:
\[ \left(\frac{x}{P}\right)^2 - \left(\frac{y}{Q}\right)^2 = 1 \]

4. **Simplifying the Equation**:
- Expanding the squares:
\[ \frac{x^2}{P^2} - \frac{y^2}{Q^2} = 1 \]
- To eliminate the fractions, we can multiply through by \( P^2 Q^2 \):
\[ Q^2 x^2 - P^2 y^2 = P^2 Q^2 \]

5. **Rearranging the Final Equation**:
- Rearranging gives us the final form:
\[ x^2 Q^2 - y^2 P^2 = P^2 Q^2 \]

Thus, the final result is:
\[ x^2 Q^2 - y^2 P^2 = P^2 Q^2 \]

### Conclusion:
The correct option is:
- \( x^2 Q^2 - y^2 P^2 = P^2 Q^2 \)