Hyperbola
Line Not Meeting Hyperbola — Discriminant Condition
nta_pyq_2026_jan
Grade 11
Question:
If the line $\alpha x+2y=1$, where $\alpha\in\mathbb{R}$, does not meet the hyperbola $x^2-9y^2=9$, then a possible value of $\alpha$ is:
Step-by-Step Solution
Key Concept: Substitute $y=\tfrac{1-\alpha x}{2}$ into $x^2-9y^2=9$: $(4-9\alpha^2)x^2+18\alpha x-45=0$. For no intersection, discriminant $<0$.
$|\alpha|>\tfrac{\sqrt{5}}{3}\approx0.745$. $\alpha=0.8$.
Correct Answer: 4