The parabolas `y=x^2-9" and "y=kx^2` intersect at points A and B. If length AB is equal to 2a, then the value of k is:

Options

A. `(a^2-9)/(a^2)`
B. `(9)/(a^2)`
C. `a^2+3`
D. None of these

Answer

To solve the problem of finding the value of \( k \) for the parabolas \( y = x^2 - 9 \) and \( y = kx^2 \) that intersect at points \( A \) and \( B \) with the length \( AB \) equal to \( 2a \), we can follow these steps:

### Step-by-Step Solution

1. **Set the equations equal to find points of intersection:**
\[ x^2 - 9 = kx^2 \]
Rearranging gives:
\[ x^2 - kx^2 - 9 = 0 \]
This simplifies to:
\[ (1 - k)x^2 - 9 = 0 \]

2. **Rearranging the equation:**
\[ (1 - k)x^2 = 9 \]
Thus,
\[ x^2 = \frac{9}{1 - k} \]

3. **Finding the x-coordinates of points A and B:**
Taking the square root gives:
\[ x = \pm \sqrt{\frac{9}{1 - k}} = \pm \frac{3}{\sqrt{1 - k}} \]
Therefore, the points of intersection are at \( x = \frac{3}{\sqrt{1 - k}} \) and \( x = -\frac{3}{\sqrt{1 - k}} \).

4. **Calculating the distance AB:**
The distance \( AB \) is given by:
\[ AB = 2x = 2 \left( \frac{3}{\sqrt{1 - k}} \right) = \frac{6}{\sqrt{1 - k}} \]

5. **Setting the distance equal to \( 2a \):**
According to the problem, we have:
\[ \frac{6}{\sqrt{1 - k}} = 2a \]

6. **Solving for \( \sqrt{1 - k} \):**
Rearranging gives:
\[ \sqrt{1 - k} = \frac{6}{2a} = \frac{3}{a} \]

7. **Squaring both sides:**
\[ 1 - k = \left( \frac{3}{a} \right)^2 \]
Thus,
\[ 1 - k = \frac{9}{a^2} \]

8. **Solving for \( k \):**
Rearranging gives:
\[ k = 1 - \frac{9}{a^2} \]

### Final Result
The value of \( k \) is:
\[ k = 1 - \frac{9}{a^2} \]