Question

If \( P(\theta) \) lies on the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and \( S \) and \( S' \) are foci of the hyperbola, then \( SP \cdot SP' = \)

• A. \( a^2 \tan^2 \theta - b^2 \sec^2 \theta \)
• B. \( a^2 \tan^2 \theta + b^2 \sec^2 \theta \)
• C. \( a^2 \sec^2 \theta + b^2 \tan^2 \theta \)
• D. \( a^2 \sec^2 \theta - b^2 \tan^2 \theta \)

Hint:

For hyperbolas, the product of distances from any point on the hyperbola to the foci has a standard form based on the angle \( \theta \).

Answer 1

The Correct Option is B

Step 1: General property of hyperbola.
For any point \( P(\theta) \) on the hyperbola, the distance from the point to the foci satisfies the equation: \[ SP \cdot SP' = a^2 \tan^2 \theta + b^2 \sec^2 \theta \] Step 2: Apply the given conditions.
Given the equation of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), and the foci at \( S \) and \( S' \), we know the relationship for the product of the distances from a point on the hyperbola to the foci is: \[ SP \cdot SP' = a^2 \tan^2 \theta + b^2 \sec^2 \theta \] Step 3: Conclusion.
Thus, the value of \( SP \cdot SP' \) is \( \boxed{a^2 \tan^2 \theta + b^2 \sec^2 \theta} \).