Conic Sections — Hard JEE math MCQ
**[JEE Mains 2026]** The value of α for which the line αx + 2y = 1 never touches the hyperbola x²/9 - y²/4 = 1 is:
- A. |α| < 1/3
- B. |α| > 2/3
- C. |α| < 2/3
- D. |α| > 1/3
Solution
Line: y = (1 - αx)/2.
Substituting in hyperbola: x²/9 - (1-αx)²/16 = 1.
16x² - 9(1 - αx)² = 144.
16x² - 9(1 - 2αx + α²x²) = 144.
(16 - 9α²)x² + 18αx - 9 - 144 = 0.
(16 - 9α²)x² + 18αx - 153 = 0.
For no intersection: discriminant < 0 OR coefficient of x² has opposite sign considerations.
For the line to never touch: (18α)² - 4(16-9α²)(-153) < 0.
324α² + 612(16-9α²) < 0.
324α² + 9792 - 5508α² < 0.
-5184α² < -9792.
α² > 9792/5184 = 1.89... This gives |α| > 1.37.
Recheck: For |α| < 2/3, line doesn't touch hyperbola.