Question

If the line \( ax + 2y = 1 \), where \( a \in \mathbb{R} \), does not meet the hyperbola \( x^2 - 9y^2 = 9 \), then a possible value of \( a \) is:

• A. 0.5
• B. 0.6
• C. 0.8
• D. 0.7

Hint:

When a line does not intersect a conic, substitute the line into the conic equation and ensure the resulting quadratic has a negative discriminant.

Answer 1

The Correct Option is D

The given hyperbola is \[ x^2 - 9y^2 = 9, \] which can be written as \[ \frac{x^2}{9} - y^2 = 1. \] The given line is \[ ax + 2y = 1. \] Step 1: Express \( y \) in terms of \( x \).
From the equation of the line, \[ 2y = 1 - ax \Rightarrow y = \frac{1 - ax}{2}. \] Step 2: Substitute into the hyperbola equation.
Substitute \( y = \frac{1 - ax}{2} \) into \( x^2 - 9y^2 = 9 \): \[ x^2 - 9\left(\frac{1 - ax}{2}\right)^2 = 9. \] \[ x^2 - \frac{9}{4}(1 - 2ax + a^2x^2) = 9. \] Multiplying throughout by 4, \[ 4x^2 - 9 + 18ax - 9a^2x^2 = 36. \] \[ (4 - 9a^2)x^2 + 18ax - 45 = 0. \] Step 3: Use the condition for no intersection.
For the line to not meet the hyperbola, the quadratic equation in \( x \) must have no real roots. Hence, its discriminant must be negative: \[ \Delta<0. \] \[ (18a)^2 - 4(4 - 9a^2)(-45)<0. \] \[ 324a^2 + 180(4 - 9a^2)<0. \] \[ 324a^2 + 720 - 1620a^2<0. \] \[ -1296a^2 + 720<0. \] \[ 1296a^2>720. \] \[ a^2>\frac{5}{9}. \] \[ |a|>\frac{\sqrt{5}}{3} \approx 0.745. \] Step 4: Choose the correct option.
Among the given options, \[ a = 0.7 \] is a valid possible value satisfying the condition. Final Answer: \[ \boxed{0.7} \]