Question

If a line \(ax + y = 1\) does not intersect the hyperbola \(x^2 - 9y^2 = 9\) then a possible value of \(\alpha\) is :

• A. 0.2
• B. 0.3
• C. 0.4
• D. 0.5

Hint:

For conic sections, knowing the conditions of tangency, intersection, and non-intersection for a line \(y = mx + c\) is a major time-saver. For a hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the condition is based on comparing \(c^2\) with \(a^2m^2 - b^2\).

Answer 1

The Correct Option is D

Step 1: Understanding the Question: 
We are given the equation of a line and a hyperbola. Our goal is to find the condition on the parameter \(a\) such that the line does not intersect the hyperbola, and then determine a possible value for \(a\) from the provided options. 

Step 2: Key Formula or Approach: 
First, we rewrite both equations in their standard forms. 
The equation of the hyperbola is \(x^2 - 9y^2 = 9\). Dividing both sides by 9, we get: \[ \frac{x^2}{9} - \frac{y^2}{1} = 1 \] This is a standard hyperbola with \(a_h^2 = 9\) and \(b_h^2 = 1\), where we use \(a_h\) and \(b_h\) to avoid confusion with the parameter \(a\) in the line's equation. 
The equation of the line is \(ax + y = 1\), which can be rewritten as: \[ y = -ax + 1 \] This is in the slope-intercept form \(y = mx + c\), where the slope \(m = -a\) and the y-intercept \(c = 1\). 
For a line \(y = mx + c\) and a hyperbola \(\frac{x^2}{a_h^2} - \frac{y^2}{b_h^2} = 1\), the condition for the line to not intersect the hyperbola is: \[ c^2<a_h^2 m^2 - b_h^2 \] (The condition for tangency is \(c^2 = a_h^2 m^2 - b_h^2\), and for intersection at two points is \(c^2>a_h^2 m^2 - b_h^2\)). 

Step 3: Detailed Explanation: 
We substitute the values from our problem into the condition for no intersection: - \(c = 1\) - \(m = -a\) - \(a_h^2 = 9\) - \(b_h^2 = 1\) 
This gives us the inequality: \[ 1^2<9(-a)^2 - 1 \] Simplifying: \[ 1<9a^2 - 1 \] \[ 2<9a^2 \] \[ a^2>\frac{2}{9} \] Taking the square root of both sides: \[ |a|>\sqrt{\frac{2}{9}} = \frac{\sqrt{2}}{3} \] Now, we approximate the decimal value of \( \frac{\sqrt{2}}{3} \): \[ \frac{\sqrt{2}}{3} \approx \frac{1.414}{3} \approx 0.471 \] So, the condition becomes: \[ |a|>0.471 \] 

Step 4: Final Answer: 
Now, we check the given options to find which one satisfies \( |a|>0.471 \): 
(A) \( |0.2| = 0.2 \), which is not greater than 0.471. 
(B) \( |0.3| = 0.3 \), which is not greater than 0.471. 
(C) \( |0.4| = 0.4 \), which is not greater than 0.471. 
(D) \( |0.5| = 0.5 \), which is greater than 0.471. 

Therefore, a possible value of \(a\) is 0.5.