Question

The value(s) of \( \alpha \) for which the line \( \alpha x + 2y = 1 \) never touches the hyperbola \[ \frac{x^2}{9} - \frac{y^2}{1} = 1 \] is/are:

• A. \( R - \sqrt{\frac{5}{2}} \)
• B. \( R - \sqrt{5}, \sqrt{5} \)
• C. \( R - \sqrt{\frac{5}{3}} \)
• D. \( R \)

Hint:

When solving geometry problems involving conic sections and lines, use the distance formula from a point to a line to determine the condition for intersection or non-intersection.

Answer 1

The Correct Option is C

Step 1: Understanding the condition for no intersection.
For the line to never touch the hyperbola, the distance between the center of the hyperbola and the line must be greater than the length of the semi-major axis of the hyperbola. Step 2: Equation of the line.
The equation of the line is \( \alpha x + 2y = 1 \). The condition for no intersection can be derived by comparing the distance of the line from the center of the hyperbola to the semi-major axis length. Step 3: Conclusion.
The value of \( \alpha \) that satisfies this condition is \( R - \sqrt{\frac{5}{3}} \). Final Answer: \[ \boxed{R - \sqrt{\frac{5}{3}}} \]