Let \( S \) be the set of positive integral values of \( a \) for which \[ \frac{a x^2 + 2(a + 1)x + 9a + 4}{x^2 - 8x + 32} < 0, \quad \forall x \in \mathbb{R}. \] Then, the number of elements in \( S \) is:
- 1
- 0
- \(\infty\)
- 3
The Correct Option is B
Approach Solution - 1
To determine the number of positive integral values of \( a \) for which the inequality
\(\frac{a x^2 + 2(a + 1)x + 9a + 4}{x^2 - 8x + 32} < 0\) holds for all \( x \in \mathbb{R} \), we need to analyze the expression carefully.
- First, observe that for the expression to be negative for all \( x \), the quadratic in the numerator must change sign such that it is less than zero when the denominator is positive. Alternatively, both the numerator and the denominator must be such that their combination makes the whole fraction negative for all \( x \).
- The denominator \( x^2 - 8x + 32 \) does not have any real roots since its discriminant \( (-8)^2 - 4 \times 1 \times 32 = 64 - 128 = -64 \) is negative. Therefore, the quadratic expression of the denominator is always positive for all real \( x \).
- For the fractional expression to be negative for all \( x \), the numerator \( a x^2 + 2(a + 1)x + 9a + 4 \) must be negative for all real \( x \).
- The discriminant of the quadratic in the numerator is given by:
- Simplifying the above discriminant:
- The numerator will always be negative for all \( x \) if its discriminant \( \Delta \) is negative, i.e.,
- Solving this inequality:
- This is a quadratic inequality whose discriminant is negative, which means there’s no real solution for \( a \) that makes it always positive for the entire domain \( \mathbb{R} \). Therefore, there is no positive integral value for \( a \) which makes the original fraction negative for all real \( x \).
Therefore, the number of elements in the set \( S \) is 0.
Approach Solution -2
Consider the inequality:
\[ ax^2 + 2(a + 1)x + 9a + 4 < 0 \quad \forall x \in \mathbb{R} \]
For the quadratic to be negative for all values of \( x \), the coefficient of \( x^2 \) must be negative:
\[ a < 0 \]
Since we are looking for positive integral values of \( a \), no such values exist.
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