A comprehensive platform for Teachers to create standard question papers and Students to practice Case-Based, Assertion-Reason, and Critical Thinking questions.
Create professional PDF/Word papers with logo, instructions, and mixed question types in minutes.
Explore our repository by Class and Topic. Filter by "Knowledge" or "Competency" levels.
For Students. Take timed MCQ tests to check your understanding. Get instant feedback.
According to NEP 2020, rote learning is out. The focus has shifted to assessing a student's ability to apply concepts in real-life situations.
Questions derived from real-world passages to test analytical skills.
Testing the logic behind concepts, not just the definition.
Open-ended scenarios that require thinking beyond the textbook.
We provide complete AI-Powered Explanations for every question.
The equation of the hyperbola is $x^2 - 9y^2 = 9$. We can rewrite this as $\frac{x^2}{9} - y^2 = 1$.
The equation of the line is $ax + y = 1$, which can be rewritten as $y = 1 - ax$.
Substitute the equation of the line into the equation of the hyperbola:
$x^2 - 9(1 - ax)^2 = 9$
$x^2 - 9(1 - 2ax + a^2x^2) = 9$
$x^2 - 9 + 18ax - 9a^2x^2 = 9$
$(1 - 9a^2)x^2 + 18ax - 18 = 0$
For the line not to intersect the hyperbola, the quadratic equation must have no real solutions. This means the discriminant must be negative.
The discriminant is $D = b^2 - 4ac = (18a)^2 - 4(1 - 9a^2)(-18)$
$D = 324a^2 + 72(1 - 9a^2) = 324a^2 + 72 - 648a^2 = 72 - 324a^2$
For no intersection, $D < 0$
$72 - 324a^2 < 0$
$324a^2 > 72$
$a^2 > \frac{72}{324} = \frac{2}{9}$
$|a| > \sqrt{\frac{2}{9}} = \frac{\sqrt{2}}{3} \approx \frac{1.414}{3} \approx 0.471$
So, $a > 0.471$ or $a < -0.471$
From the options, the possible value of $a$ is 0.5.
Correct Answer: 0.5