If a line \(ax + y = 1\) does not intersect the hyperbola \(x^2 - 9y^2 = 9\) then a possible value of \(\alpha\) is :
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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 \(a^2m^2 - b^2\).
Step 1: Understanding the Question:
We are given the equation of a line and a hyperbola. We need to find the condition on the parameter 'a' such that the line does not intersect the hyperbola, and then identify a possible value for 'a' from the options.
Step 2: Key Formula or Approach:
First, we write both equations in their standard forms.
The equation of the hyperbola is \(x^2 - 9y^2 = 9\). Dividing by 9, we get:
\[ \frac{x^2}{9} - \frac{y^2}{1} = 1 \]
This is a standard hyperbola with \(a_h^2 = 9\) and \(b_h^2 = 1\). (Using \(a_h, b_h\) to avoid confusion with parameter 'a' in the line).
The equation of the line is \(ax + y = 1\), which can be written as \(y = -ax + 1\).
This is in the slope-intercept form \(y = mx + c\), with slope \(m = -a\) and 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 \(c^2<a_h^2m^2 - b_h^2\).
(The condition for tangency is \(c^2 = a_h^2m^2 - b_h^2\), and for intersection at two points is \(c^2>a_h^2m^2 - b_h^2\)).
Step 3: Detailed Explanation:
Substitute the values from our problem into the condition for no intersection:
\(c=1\), \(m=-a\), \(a_h^2 = 9\), \(b_h^2 = 1\).
\[ 1^2<9(-a)^2 - 1 \]
\[ 1<9a^2 - 1 \]
\[ 2<9a^2 \]
\[ a^2>\frac{2}{9} \]
Taking the square root of both sides:
\[ |a|>\sqrt{\frac{2}{9}} = \frac{\sqrt{2}}{3} \]
Now, we need to find an approximate decimal value for \(\frac{\sqrt{2}}{3}\):
\[ \frac{\sqrt{2}}{3} \approx \frac{1.414}{3} \approx 0.471 \]
So, the condition is \(|a|>0.471\).
Step 4: Final Answer:
We check the given options to see 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.
