If `M` and `m` denotes maximum and minimum value of `sqrt(49cos^2theta + sin^2theta) + sqrt(49sin^2theta+cos^2theta` then find the value of `(M+m)` is

Answer

To solve the problem, we need to find the maximum (M) and minimum (m) values of the expression:

\[ u = \sqrt{49 \cos^2 \theta + \sin^2 \theta} + \sqrt{49 \sin^2 \theta + \cos^2 \theta} \]

### Step 1: Rewrite the expression

We can rewrite the expression for \( u \) as follows:

\[ u = \sqrt{48 \cos^2 \theta + 1} + \sqrt{48 \sin^2 \theta + 1} \]

### Step 2: Square both sides

To simplify the calculations, we will square \( u \):

\[ u^2 = \left(\sqrt{48 \cos^2 \theta + 1} + \sqrt{48 \sin^2 \theta + 1}\right)^2 \]

### Step 3: Expand the squared expression

Using the formula \( (a + b)^2 = a^2 + b^2 + 2ab \):

\[ u^2 = (48 \cos^2 \theta + 1) + (48 \sin^2 \theta + 1) + 2\sqrt{(48 \cos^2 \theta + 1)(48 \sin^2 \theta + 1)} \]

### Step 4: Combine terms

Combine the terms:

\[ u^2 = 48 (\cos^2 \theta + \sin^2 \theta) + 2 + 2\sqrt{(48 \cos^2 \theta + 1)(48 \sin^2 \theta + 1)} \]

Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \):

\[ u^2 = 48 + 2 + 2\sqrt{(48 \cos^2 \theta + 1)(48 \sin^2 \theta + 1)} \]

### Step 5: Simplify further

This gives us:

\[ u^2 = 50 + 2\sqrt{(48 \cos^2 \theta + 1)(48 \sin^2 \theta + 1)} \]

### Step 6: Find the maximum and minimum values

To find the maximum and minimum values of \( u \), we need to analyze the term \( \sqrt{(48 \cos^2 \theta + 1)(48 \sin^2 \theta + 1)} \).

1. **Maximum value of \( \sin^2 2\theta \)**: The maximum value of \( \sin^2 2\theta \) is 1. Thus:

\[ \sqrt{(48 \cdot 1 + 1)(48 \cdot 0 + 1)} = \sqrt{49} = 7 \]

So, the maximum value \( M \) occurs when \( \sin^2 2\theta = 1 \):

\[ u = \sqrt{50 + 2 \cdot 7} = \sqrt{64} = 8 \]

2. **Minimum value of \( \sin^2 2\theta \)**: The minimum value of \( \sin^2 2\theta \) is 0. Thus:

\[ \sqrt{(48 \cdot 0 + 1)(48 \cdot 1 + 1)} = \sqrt{1 \cdot 49} = 7 \]

So, the minimum value \( m \) occurs when \( \sin^2 2\theta = 0 \):

\[ u = \sqrt{50 + 0} = \sqrt{50} = 7 \]

### Step 7: Calculate \( M + m \)

Now, we can find \( M + m \):

\[ M + m = 10 + 8 = 18 \]

### Final Answer

Thus, the value of \( M + m \) is:

\[ \boxed{18} \]