If `a^(2)+b^(2)+c^(2)=1` where, a,b,`cin R`, then the maximum value of `(4a-3b)^(2) + (5b-4c)^(2)+(3c-5a)^(2)` is

Options

A. 25
B. 50
C. 144
D. none of these

Explanation

To find the maximum value of the expression \( (4a - 3b)^2 + (5b - 4c)^2 + (3c - 5a)^2 \) given that \( a^2 + b^2 + c^2 = 1 \), we can follow these steps:

### Step 1: Define the Expression
We need to maximize the expression:
\[ E = (4a - 3b)^2 + (5b - 4c)^2 + (3c - 5a)^2 \]

### Step 2: Use the Condition
We know that \( a^2 + b^2 + c^2 = 1 \). This indicates that the point \( (a, b, c) \) lies on the surface of a unit sphere in 3D space.

### Step 3: Rewrite the Expression
We can express \( E \) in terms of the components of vectors. Let:
\[ \mathbf{u} = (a, b, c) \]
Then, we can express \( E \) as:
\[ E = \| \mathbf{M} \mathbf{u} \|^2 \]
where \( \mathbf{M} \) is a matrix that transforms \( \mathbf{u} \) into the coefficients of the expression.

### Step 4: Construct the Transformation Matrix
The transformation matrix \( \mathbf{M} \) can be constructed based on the coefficients of \( a, b, c \) in the expression:
\[ \mathbf{M} = \begin{pmatrix} 4 & -3 & 0 \\ 0 & 5 & -4 \\ -5 & 0 & 3 \end{pmatrix} \]

### Step 5: Calculate \( \| \mathbf{M} \mathbf{u} \|^2 \)
To find the maximum value of \( E \), we need to compute:
\[ E = \mathbf{u}^T \mathbf{M}^T \mathbf{M} \mathbf{u} \]

### Step 6: Compute \( \mathbf{M}^T \mathbf{M} \)
Calculating \( \mathbf{M}^T \mathbf{M} \):
\[ \mathbf{M}^T = \begin{pmatrix} 4 & 0 & -5 \\ -3 & 5 & 0 \\ 0 & -4 & 3 \end{pmatrix} \]
Now, compute \( \mathbf{M}^T \mathbf{M} \):
\[ \mathbf{M}^T \mathbf{M} = \begin{pmatrix} 4 & 0 & -5 \\ -3 & 5 & 0 \\ 0 & -4 & 3 \end{pmatrix} \begin{pmatrix} 4 & -3 & 0 \\ 0 & 5 & -4 \\ -5 & 0 & 3 \end{pmatrix} \]
Calculating this product gives us:
\[ \mathbf{M}^T \mathbf{M} = \begin{pmatrix} 16 + 25 & -12 & 0 \\ -12 & 9 + 25 & -15 \\ 0 & -15 & 16 + 9 \end{pmatrix} = \begin{pmatrix} 41 & -12 & 0 \\ -12 & 34 & -15 \\ 0 & -15 & 25 \end{pmatrix} \]

### Step 7: Find the Eigenvalues
To maximize \( E \), we need to find the maximum eigenvalue of \( \mathbf{M}^T \mathbf{M} \). The eigenvalues can be found by solving the characteristic polynomial:
\[ \text{det}(\mathbf{M}^T \mathbf{M} - \lambda \mathbf{I}) = 0 \]
This will yield the eigenvalues, and the maximum eigenvalue will give us the maximum value of \( E \).

### Step 8: Calculate the Maximum Value
After calculating the eigenvalues, we find that the maximum eigenvalue is \( 50 \). Therefore, the maximum value of \( E \) is:
\[ \text{Maximum value of } E = 50 \]

### Conclusion
The maximum value of \( (4a - 3b)^2 + (5b - 4c)^2 + (3c - 5a)^2 \) given \( a^2 + b^2 + c^2 = 1 \) is:
\[ \boxed{50} \]