If the line , where , does not meet the hyperbola x2-9y2=9 then a possible value of is: [2026]

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Solution

Q.
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: [2026]

1 0.5

2 0.6

3 0.7

4 0.8

Ans.
(4)

$y = \frac{1 - α x}{2}$

Put this in equation of hyperbola

$∴$ $x^{2} - 9 {(\frac{1 - α x}{2})}^{2} = 9$

$(4 - 9 α^{2}) x^{2} + 18 α x - 45 = 0$

$∵$ $line does not intersect hyperbola$

$∴$ $D < 0$

$\Rightarrow α^{2} - \frac{5}{9} > 0$

$\Rightarrow α \in (- \infty, - \frac{\sqrt{5}}{3}) \cup (\frac{\sqrt{5}}{3}, \infty)$

$Here \frac{\sqrt{5}}{3} \approx 0.74$

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