If `2x ^(2) +5x+5x +7 =0 and ax ^(2)+ +bx+c=0` have at least one root common such that `a,b,c in {1,2,……,100},` then the difference between the maximum and minimum vlaues of `a +b +c` is:

Options

A. 196
B. 284
C. 182
D. 126

Answer

To solve the problem step by step, we will analyze the given quadratic equations and find the required values of \(a\), \(b\), and \(c\).

### Step 1: Understand the given equations
We have two quadratic equations:
1. \(2x^2 + 5x + 7 = 0\)
2. \(ax^2 + bx + c = 0\)

These equations have at least one root in common.

### Step 2: Set up the ratio of coefficients
For two quadratic equations to have a common root, the ratios of their coefficients must be equal. Thus, we can write:
\[ \frac{a}{2} = \frac{b}{5} = \frac{c}{7} = k \]
where \(k\) is some constant.

### Step 3: Express \(a\), \(b\), and \(c\) in terms of \(k\)
From the ratios, we can express \(a\), \(b\), and \(c\) as:
\[ a = 2k, \quad b = 5k, \quad c = 7k \]

### Step 4: Find \(a + b + c\)
Now, we can find the sum \(a + b + c\):
\[ a + b + c = 2k + 5k + 7k = 14k \]

### Step 5: Determine the range for \(k\)
Given that \(a\), \(b\), and \(c\) must be integers in the set \(\{1, 2, \ldots, 100\}\), we need to find the possible values of \(k\) such that \(a\), \(b\), and \(c\) remain within this range.

1. For \(a = 2k\):
- Minimum: \(2k \geq 1 \Rightarrow k \geq 0.5\)
- Maximum: \(2k \leq 100 \Rightarrow k \leq 50\)

2. For \(b = 5k\):
- Minimum: \(5k \geq 1 \Rightarrow k \geq 0.2\)
- Maximum: \(5k \leq 100 \Rightarrow k \leq 20\)

3. For \(c = 7k\):
- Minimum: \(7k \geq 1 \Rightarrow k \geq \frac{1}{7} \approx 0.142857\)
- Maximum: \(7k \leq 100 \Rightarrow k \leq \frac{100}{7} \approx 14.2857\)

### Step 6: Find the feasible range for \(k\)
The most restrictive limits for \(k\) come from \(b\) and \(c\):
- Minimum \(k \geq 0.5\)
- Maximum \(k \leq 20\)

Thus, the feasible range for \(k\) is:
\[ 0.5 \leq k \leq 14 \]

### Step 7: Calculate minimum and maximum values of \(a + b + c\)
1. **Minimum value of \(a + b + c\)**:
- When \(k = 1\):
\[ a + b + c = 14 \times 1 = 14 \]

2. **Maximum value of \(a + b + c\)**:
- When \(k = 14\):
\[ a + b + c = 14 \times 14 = 196 \]

### Step 8: Find the difference between maximum and minimum values
Now, we can find the difference:
\[ \text{Difference} = 196 - 14 = 182 \]

### Final Answer
The difference between the maximum and minimum values of \(a + b + c\) is:
\[ \boxed{182} \]