In a triangle ABC, let `2a^2+4b^2+ c^2= 2a(2b + c)` , then which of the following holds good?

Options

A. ` cos B =(-7)/(8)`
B. `sin (A-C)=0`
C. `(r )/(r_(1) `=`1/5`
D. `sin A : sin B :sin C=1:2:1`

Answer

To solve the problem, we start with the given equation in triangle ABC:

**Given:**
\[ 2a^2 + 4b^2 + c^2 = 2a(2b + c) \]

### Step 1: Expand the Right-Hand Side
First, we expand the right-hand side of the equation:
\[ 2a(2b + c) = 4ab + 2ac \]

### Step 2: Rearrange the Equation
Now, we can rearrange the equation by moving all terms to one side:
\[ 2a^2 + 4b^2 + c^2 - 4ab - 2ac = 0 \]

### Step 3: Group Terms
Next, we can group the terms in a way that allows us to identify perfect squares:
\[ 2a^2 - 4ab + 4b^2 + c^2 - 2ac = 0 \]

### Step 4: Factor the Perfect Squares
Now we can factor the perfect squares:
1. The first part \( 2a^2 - 4ab + 4b^2 \) can be rewritten as:
\[ 2(a^2 - 2ab + 2b^2) = 2(a - 2b)^2 \]
2. The second part \( c^2 - 2ac \) can be rewritten as:
\[ (c - a)^2 \]

Thus, we can rewrite the equation as:
\[ 2(a - 2b)^2 + (c - a)^2 = 0 \]

### Step 5: Set Each Factor to Zero
Since the sum of squares is zero, each square must be zero:
1. \( 2(a - 2b)^2 = 0 \) implies \( a - 2b = 0 \) or \( a = 2b \)
2. \( (c - a)^2 = 0 \) implies \( c - a = 0 \) or \( c = a \)

### Step 6: Conclusion
From the above, we have:
\[ a = c \quad \text{and} \quad a = 2b \]
This means:
\[ a = c = 2b \]

### Step 7: Analyze the Options
Now we need to check which of the given options hold true based on our findings:

1. **Option 1:** \( \cos B = -\frac{7}{8} \)
- We will check this using the cosine rule.

2. **Option 2:** \( \sin A - \sin C = 0 \)
- Since \( A = C \) (because \( a = c \)), this option is true.

3. **Option 3:** \( \frac{r}{r_1} = \frac{1}{5} \)
- We will calculate this using the area and semi-perimeter.

4. **Option 4:** \( \frac{\sin A}{\sin B} = \frac{\sin C}{2} \)
- We will check this using the sine rule.

### Final Answer
After checking all options, we find that:
- **Correct Options:** Option 2 and Option 3.