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 intersection of the parabolas \( y = x^2 - 9 \) and \( y = kx^2 \) such that the length of segment \( AB \) is equal to \( 2a \), we can follow these steps:

### Step 1: Set the equations equal to find points of intersection
We start by setting the two equations equal to each other to find the points of intersection:
\[ x^2 - 9 = kx^2 \]
Rearranging gives:
\[ x^2 - kx^2 = 9 \]
Factoring out \( x^2 \):
\[ (1 - k)x^2 = 9 \]

### Step 2: Solve for \( x^2 \)
Now, we can solve for \( x^2 \):
\[ x^2 = \frac{9}{1 - k} \]

### Step 3: Find the values of \( x \)
Taking the square root gives us the values of \( x \):
\[ x = \pm \sqrt{\frac{9}{1 - k}} = \pm \frac{3}{\sqrt{1 - k}} \]

### Step 4: Determine the length of segment \( AB \)
The length of segment \( AB \) is the distance between the two intersection points, which is given by:
\[ AB = |x_1 - x_2| = \left| \frac{3}{\sqrt{1 - k}} - \left(-\frac{3}{\sqrt{1 - k}}\right) \right| = \frac{6}{\sqrt{1 - k}} \]

### Step 5: Set the length equal to \( 2a \)
According to the problem, we have:
\[ \frac{6}{\sqrt{1 - k}} = 2a \]
Dividing both sides by 2 gives:
\[ \frac{3}{\sqrt{1 - k}} = a \]

### Step 6: Square both sides to eliminate the square root
Now we square both sides:
\[ \left(\frac{3}{\sqrt{1 - k}}\right)^2 = a^2 \]
This simplifies to:
\[ \frac{9}{1 - k} = a^2 \]

### Step 7: Rearranging to find \( k \)
Now, we can rearrange this equation to solve for \( k \):
\[ 9 = a^2(1 - k) \]
Expanding gives:
\[ 9 = a^2 - a^2k \]
Rearranging for \( k \):
\[ a^2k = a^2 - 9 \]
Thus:
\[ k = \frac{a^2 - 9}{a^2} \]

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