The parabolas `y=x^2-9` and `y=lamdax^2` intersect at points A and B. If length of AB is equal to 2a and if `lamda a^2+mu=a^2`, then the value of `mu` is

Answer

To solve the problem, we need to find the value of \( \mu \) given the equations of two parabolas and the conditions provided. Let's go through the solution step by step.

### Step 1: Set up the equations of the parabolas
The equations of the parabolas are:
1. \( y = x^2 - 9 \) (Equation 1)
2. \( y = \lambda x^2 \) (Equation 2)

### Step 2: Find the points of intersection
To find the points of intersection, we equate the two equations:
\[ x^2 - 9 = \lambda x^2 \]
Rearranging gives:
\[ x^2(1 - \lambda) = 9 \]
Thus, we can solve for \( x^2 \):
\[ x^2 = \frac{9}{1 - \lambda} \]
Taking the square root gives:
\[ x = \pm \frac{3}{\sqrt{1 - \lambda}} \]

### Step 3: Calculate the corresponding \( y \)-coordinates
Substituting \( x \) back into either equation (let's use Equation 2) to find \( y \):
\[ y = \lambda x^2 = \lambda \left(\frac{9}{1 - \lambda}\right) = \frac{9\lambda}{1 - \lambda} \]
Thus, the points of intersection \( A \) and \( B \) are:
- \( A\left(\frac{3}{\sqrt{1 - \lambda}}, \frac{9\lambda}{1 - \lambda}\right) \)
- \( B\left(-\frac{3}{\sqrt{1 - \lambda}}, \frac{9\lambda}{1 - \lambda}\right) \)

### Step 4: Find the distance \( AB \)
The distance \( AB \) can be calculated using the distance formula. Since both points have the same \( y \)-coordinate, the distance simplifies to:
\[ AB = x_2 - x_1 = \left(-\frac{3}{\sqrt{1 - \lambda}} - \frac{3}{\sqrt{1 - \lambda}}\right) = \frac{6}{\sqrt{1 - \lambda}} \]

### Step 5: Set the distance equal to \( 2a \)
According to the problem, the length of \( AB \) is equal to \( 2a \):
\[ \frac{6}{\sqrt{1 - \lambda}} = 2a \]
Dividing both sides by 2 gives:
\[ \frac{3}{\sqrt{1 - \lambda}} = a \]
Squaring both sides results in:
\[ a^2 = \frac{9}{1 - \lambda} \]

### Step 6: Substitute into the given condition
We are given the condition:
\[ \lambda a^2 + \mu = a^2 \]
Substituting \( a^2 = \frac{9}{1 - \lambda} \) into this equation:
\[ \lambda \left(\frac{9}{1 - \lambda}\right) + \mu = \frac{9}{1 - \lambda} \]
Rearranging gives:
\[ \mu = \frac{9}{1 - \lambda} - \frac{9\lambda}{1 - \lambda} \]
This simplifies to:
\[ \mu = \frac{9(1 - \lambda)}{1 - \lambda} = 9 \]

### Final Answer
Thus, the value of \( \mu \) is:
\[ \mu = 9 \]