If `u = sqrt( a^(2) cos^(2) theta + b^(2) sin^(2) theta ) + sqrt( a^(2) sin^(2) theta + b^(2) cos ^(2) theta )` then the difference between maximum and minimum values of `u^(2)` is given by

Options

A. `2( a^(2) + b^(2))`
B. `2 sqrt( a^(2) + b^(2))`
C. `( a+b)^(2)`
D. `( a- b)^(2)`

Answer

To find the difference between the maximum and minimum values of \( u^2 \) given the expression

\[ u = \sqrt{a^2 \cos^2 \theta + b^2 \sin^2 \theta} + \sqrt{a^2 \sin^2 \theta + b^2 \cos^2 \theta}, \]

we can follow these steps:

### Step 1: Square the expression for \( u \)

First, we square \( u \):

\[ u^2 = \left( \sqrt{a^2 \cos^2 \theta + b^2 \sin^2 \theta} + \sqrt{a^2 \sin^2 \theta + b^2 \cos^2 \theta} \right)^2. \]

Using the identity \( (x + y)^2 = x^2 + y^2 + 2xy \):

\[ u^2 = \left( a^2 \cos^2 \theta + b^2 \sin^2 \theta \right) + \left( a^2 \sin^2 \theta + b^2 \cos^2 \theta \right) + 2 \sqrt{(a^2 \cos^2 \theta + b^2 \sin^2 \theta)(a^2 \sin^2 \theta + b^2 \cos^2 \theta)}. \]

### Step 2: Simplify the expression

Combining the first two terms:

\[ u^2 = a^2 (\cos^2 \theta + \sin^2 \theta) + b^2 (\sin^2 \theta + \cos^2 \theta) + 2 \sqrt{(a^2 \cos^2 \theta + b^2 \sin^2 \theta)(a^2 \sin^2 \theta + b^2 \cos^2 \theta)}. \]

Since \( \cos^2 \theta + \sin^2 \theta = 1 \):

\[ u^2 = a^2 + b^2 + 2 \sqrt{(a^2 \cos^2 \theta + b^2 \sin^2 \theta)(a^2 \sin^2 \theta + b^2 \cos^2 \theta)}. \]

### Step 3: Analyze the square root term

Let \( x = a^2 \cos^2 \theta + b^2 \sin^2 \theta \) and \( y = a^2 \sin^2 \theta + b^2 \cos^2 \theta \). The product \( xy \) can be expressed as:

\[ xy = (a^2 \cos^2 \theta + b^2 \sin^2 \theta)(a^2 \sin^2 \theta + b^2 \cos^2 \theta). \]

### Step 4: Find maximum and minimum values

To find the maximum and minimum values of \( u^2 \), we can analyze the behavior of \( u^2 \) as \( \theta \) varies.

1. **Maximum value** occurs when \( \theta = \frac{\pi}{4} \):
\[ u^2_{\text{max}} = 2(a^2 + b^2). \]

2. **Minimum value** occurs when \( \theta = 0 \):
\[ u^2_{\text{min}} = (a + b)^2. \]

### Step 5: Calculate the difference

Now, we find the difference between the maximum and minimum values:

\[ \text{Difference} = u^2_{\text{max}} - u^2_{\text{min}} = [2(a^2 + b^2)] - [(a + b)^2]. \]

Expanding \( (a + b)^2 \):

\[ (a + b)^2 = a^2 + 2ab + b^2. \]

Thus, the difference becomes:

\[ \text{Difference} = 2(a^2 + b^2) - (a^2 + 2ab + b^2) = (2a^2 + 2b^2 - a^2 - 2ab - b^2) = (a^2 - 2ab + b^2) = (a - b)^2. \]

### Final Answer

The difference between the maximum and minimum values of \( u^2 \) is:

\[ \boxed{(a - b)^2}. \]