Conic Sections
Conic Section
star_batch_jee_advanced_2025
Grade None
Question:
If a ray of light incident along the line $3x + (5 - 4\sqrt{2})y = 15$ gets reflected from the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$ at the point $(4\sqrt{2}, 3)$, then its reflected ray goes along the line:
$\sqrt{2}x - y + 5 = 0$
$\sqrt{2}y - x + 5 = 0$
$\sqrt{2}y - x - 5 = 0$
None of these
Step-by-Step Solution
Key Concept: The reflection property of hyperbolas states that a ray directed toward one focus reflects toward the other focus.
The hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$ has $e = \frac{5}{4}$ and foci at $(\pm 5, 0)$. The incident ray passes through $P(4\sqrt{2}, 3)$ and reflects off the hyperbola through $(5, 0)$. Using the reflection property, the reflected ray must pass through $(-5, 0)$, giving equation $3x - y(4\sqrt{2} + 5) + 15 = 0$.
Correct Answer: 4