Equation of two S.H.M. `x_(1)=5 sin (2pi t+pi//4),x_(2)=5sqrt(2)(sin 2pit+cos 2pit)` ratio of amplitude & phase difference will be

A. `2:1,0`
B. `1:2,0`
C. `1:2,pi//2`
D. `2:1,pi//2`

Options

A. `2:1,0`
B. `1:2,0`
C. `1:2,pi//2`
D. `2:1,pi//2`

Explanation

To solve the problem, we need to analyze the two given equations of Simple Harmonic Motion (SHM) and extract the amplitude and phase difference from them.

### Step 1: Identify the first SHM equation
The first SHM equation is given as:
\[ x_1 = 5 \sin(2\pi t + \frac{\pi}{4}) \]
From this equation, we can directly identify:
- Amplitude \( A_1 = 5 \)
- Phase angle \( \phi_1 = \frac{\pi}{4} \)

### Step 2: Identify the second SHM equation
The second SHM equation is given as:
\[ x_2 = 5\sqrt{2} \left( \sin(2\pi t) + \cos(2\pi t) \right) \]
To express this in the standard SHM form \( A \sin(2\pi t + \phi) \), we can use the trigonometric identity:
\[ \sin(a) + \cos(a) = \sqrt{2} \sin\left(a + \frac{\pi}{4}\right) \]
Thus, we can rewrite \( x_2 \):
\[ x_2 = 5\sqrt{2} \cdot \sqrt{2} \sin\left(2\pi t + \frac{\pi}{4}\right) = 10 \sin\left(2\pi t + \frac{\pi}{4}\right) \]
From this equation, we can identify:
- Amplitude \( A_2 = 10 \)
- Phase angle \( \phi_2 = \frac{\pi}{4} \)

### Step 3: Calculate the ratio of amplitudes
Now we can find the ratio of the amplitudes:
\[ \text{Ratio of amplitudes} = \frac{A_1}{A_2} = \frac{5}{10} = \frac{1}{2} \]
So, the ratio of amplitudes is \( 1:2 \).

### Step 4: Calculate the phase difference
Next, we calculate the phase difference:
\[ \text{Phase difference} = \phi_1 - \phi_2 = \frac{\pi}{4} - \frac{\pi}{4} = 0 \]

### Conclusion
Based on the calculations:
- The ratio of amplitudes is \( 1:2 \)
- The phase difference is \( 0 \)

Thus, the correct answer is:
**B. \( 1:2, 0 \)**

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