Step 1: Recall the properties of equivalence relations: a relation \( R \) is an equivalence relation if it is reflexive, symmetric, and transitive.
Step 2: - Reflexive: For every \( x \) in the given interval, \( x R x \) must hold. That is, we check if \( \sec^2 x - \tan^2 x = 1 \). This is true for all \( x \) in the interval \( \left[ 0, \frac{\pi}{2} \right] \), so the relation is reflexive.
- Symmetric: For the relation to be symmetric, if \( x R y \), then \( y R x \) must also hold. Since the equation involves both \( x \) and \( y \) in a symmetric manner, the relation is symmetric.
- Transitive: For transitivity, if \( x R y \) and \( y R z \), then \( x R z \) must hold. This property holds as well, meaning the relation is transitive. Thus, \( R \) is reflexive, symmetric, and transitive, so it is an equivalence relation.
Let one focus of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be at $ (\sqrt{10}, 0) $, and the corresponding directrix be $ x = \frac{\sqrt{10}}{2} $. If $ e $ and $ l $ are the eccentricity and the latus rectum respectively, then $ 9(e^2 + l) $ is equal to:
The largest $ n \in \mathbb{N} $ such that $ 3^n $ divides 50! is: