`x^2 +y^2= 9 & 4a^2 + 9b^2= 16`, then maximum value of `4a^2 x^2+ 9b^2y^2- 12abxy` is -

Options

A. 81
B. 100
C. 121
D. 144

Answer

To find the maximum value of the expression \( 4a^2 x^2 + 9b^2 y^2 - 12abxy \) given the constraints \( x^2 + y^2 = 9 \) and \( 4a^2 + 9b^2 = 16 \), we can follow these steps:

### Step 1: Parametrize the variables
Given the equation \( x^2 + y^2 = 9 \), we can parametrize \( x \) and \( y \) as:
\[ x = 3 \cos \theta, \quad y = 3 \sin \theta \]

### Step 2: Parametrize \( a \) and \( b \)
From the equation \( 4a^2 + 9b^2 = 16 \), we can parametrize \( a \) and \( b \) as:
\[ a = 2 \cos \phi, \quad b = \frac{4}{3} \sin \phi \]

### Step 3: Substitute into the expression
Now, substitute \( x \), \( y \), \( a \), and \( b \) into the expression:
\[ 4a^2 x^2 + 9b^2 y^2 - 12abxy \]
Substituting the values:
\[ = 4(2 \cos \phi)^2 (3 \cos \theta)^2 + 9\left(\frac{4}{3} \sin \phi\right)^2 (3 \sin \theta)^2 - 12(2 \cos \phi)\left(\frac{4}{3} \sin \phi\right)(3 \cos \theta)(3 \sin \theta) \]

### Step 4: Simplify the expression
Calculating each term:
1. \( 4(2 \cos \phi)^2 (3 \cos \theta)^2 = 4 \cdot 4 \cos^2 \phi \cdot 9 \cos^2 \theta = 144 \cos^2 \phi \cos^2 \theta \)
2. \( 9\left(\frac{4}{3} \sin \phi\right)^2 (3 \sin \theta)^2 = 9 \cdot \frac{16}{9} \sin^2 \phi \cdot 9 \sin^2 \theta = 144 \sin^2 \phi \sin^2 \theta \)
3. \( -12(2 \cos \phi)\left(\frac{4}{3} \sin \phi\right)(3 \cos \theta)(3 \sin \theta) = -12 \cdot 2 \cdot \frac{4}{3} \cdot 9 \cos \phi \sin \phi \cos \theta \sin \theta = -96 \cos \phi \sin \phi \cos \theta \sin \theta \)

Combining these:
\[ = 144 (\cos^2 \phi \cos^2 \theta + \sin^2 \phi \sin^2 \theta - \frac{2}{3} \cos \phi \sin \phi \cos \theta \sin \theta) \]

### Step 5: Use trigonometric identities
Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ = 144 \left( \frac{1}{2} + \frac{1}{2} \sin 2\phi \sin 2\theta \right) \]

### Step 6: Find the maximum value
The maximum value of \( \sin 2\phi \sin 2\theta \) is 1, thus:
\[ = 144 \left( \frac{1}{2} + \frac{1}{2} \cdot 1 \right) = 144 \cdot 1 = 144 \]

### Final Answer
The maximum value of the expression \( 4a^2 x^2 + 9b^2 y^2 - 12abxy \) is:
\[ \boxed{144} \]