Number of A parabola of the form `y= ax^2 +bx+c` with `a>0` intersection (s)of these graph of `f(x)=1/(x^2-4)` .number of a possible distinct intersection(s) of these graph is

Options

A. 0
B. 2
C. 3
D. 4

Answer

To find the number of possible distinct intersections between the parabola \( y = ax^2 + bx + c \) (where \( a > 0 \)) and the function \( f(x) = \frac{1}{x^2 - 4} \), we will analyze the behavior of both graphs.

### Step-by-Step Solution:

1. **Identify the Function Behavior:**
The function \( f(x) = \frac{1}{x^2 - 4} \) has vertical asymptotes at \( x = 2 \) and \( x = -2 \) because the denominator becomes zero at these points. The function approaches infinity as \( x \) approaches \( 2 \) or \( -2 \) from either side.

2. **Determine the Values of \( f(x) \):**
- For \( x < -2 \): \( f(x) \) is positive and approaches \( 0 \) as \( x \) decreases.
- For \( -2 < x < 2 \): \( f(x) \) is negative and approaches \( -\infty \) as \( x \) approaches \( -2 \) and \( 2 \).
- For \( x > 2 \): \( f(x) \) is again positive and approaches \( 0 \) as \( x \) increases.

3. **Analyze the Parabola:**
The parabola \( y = ax^2 + bx + c \) opens upwards since \( a > 0 \). The vertex of the parabola will determine its minimum point.

4. **Finding Intersection Points:**
To find the intersections, set the equations equal to each other:
\[ ax^2 + bx + c = \frac{1}{x^2 - 4} \]
Rearranging gives:
\[ ax^2 + bx + c - \frac{1}{x^2 - 4} = 0 \]
This is a rational equation that can be analyzed further.

5. **Behavior Near Asymptotes:**
- As \( x \) approaches \( 2 \) or \( -2 \), \( f(x) \) approaches \( \infty \) or \( -\infty \). The parabola can intersect \( f(x) \) in different ways depending on its position relative to the asymptotes.
- If the vertex of the parabola is above the horizontal line \( y = 0 \), it can intersect \( f(x) \) twice (once on each side of the asymptotes).
- If the vertex is below \( y = 0 \), it can intersect \( f(x) \) three times (once in the middle and once on each side) or four times if it crosses the horizontal line twice before and after the asymptotes.

6. **Conclusion:**
Based on the analysis, the parabola can intersect the function \( f(x) \) at:
- 2 points (if the parabola is above the horizontal line and does not touch the negative region),
- 3 points (if it dips into the negative region),
- 4 points (if it crosses the horizontal line twice).

Thus, the possible number of distinct intersections is **2, 3, or 4**.