JEE Mathematics 2026 Question with Solution

Given: The line is $ax + 2y = 1$ and the hyperbola is $x^2 - 9y^2 = 9$.

Find: A possible value of $a$ such that the line does not meet the hyperbola.

Write the hyperbola as

$$\frac{x^2}{9} - y^2 = 1$$

From the line,

$$2y = 1 - ax \Rightarrow y = \frac{1 - ax}{2}$$

Substitute this into $x^2 - 9y^2 = 9$:

$$x^2 - 9\left(\frac{1 - ax}{2}\right)^2 = 9$$ $$x^2 - \frac{9}{4}(1 - 2ax + a^2x^2) = 9$$

Multiplying throughout by $4$,

$$4x^2 - 9 + 18ax - 9a^2x^2 = 36$$ $$(4 - 9a^2)x^2 + 18ax - 45 = 0$$

For the line to not meet the hyperbola, this quadratic in $x$ must have no real roots. Therefore, its discriminant must be negative:

$$\Delta < 0$$ $$(18a)^2 - 4(4 - 9a^2)(-45) < 0$$ $$324a^2 + 180(4 - 9a^2) < 0$$ $$324a^2 + 720 - 1620a^2 < 0$$ $$-1296a^2 + 720 < 0$$ $$1296a^2 > 720$$ $$a^2 > \frac{5}{9}$$ $$|a| > \frac{\sqrt{5}}{3} \approx 0.745$$

Among the given options, the solution states that $a = 0.7$ is the valid possible value. Therefore, the correct option is D.

The solution derives

$$|a| > \frac{\sqrt{5}}{3} \approx 0.745$$

This inequality is satisfied by $0.8$ but not by $0.7$. So the algebra shown in the solution points to option C, while the solution's explicitly marks option D and concludes with $0.7$. Following the solution for extraction, the recorded answer is D, but there is a clear discrepancy between the working and the stated final answer.