If the line $\alpha x+2 y=1$, where $\alpha \in \mathbb{R}$, does not meet the hyperbola $x^{2}-9 y^{2}=9$, then a possible value of $\alpha$ is:

JEE Main · 2026 · 22 Jan 2026 · Shift 1 · Hyperbola

If the line

α x + 2 y = 1

, where

α \in R

, does not meet the hyperbola

x^{2} - 9 y^{2} = 9

, then a possible value of

α

is:

(A)0.6
(B)0.8✓ Correct
(C)0.5
(D)0.7

Answer: (B)

Solution

Substituting

y = \frac{1 - α x}{2}

into

x^{2} - 9 y^{2} = 9

gives

(4 - 9 α^{2}) x^{2} + 18 α x - 45 = 0

. The line misses the hyperbola when this equation has no real root, i.e. when its discriminant is negative:

(18 α)^{2} + 180 (4 - 9 α^{2}) < 0

, which simplifies to

720 - 1296 α^{2} < 0

, hence

α^{2} > \frac{5}{9}

, i.e.

∣ α ∣ > \frac{5}{3} \approx 0.745

. A value of

α

satisfying this, so that the line does not meet the hyperbola, is

0.8

JEE Main · 2026 · 22 Jan 2026 · Shift 1 · Single correct option

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