Let `a_m (m = 1, 2, ,p)` be the possible integral values of a for which the graphs of `f(x) =ax^2+2bx +b` and `g(x) =5x^2-3bx-a`meets at some point for all real values of b Let `t_r = prod_(m=1)^p(r-a_m )` and `S_n =sum_(r=1)^n t_r. n in N` The minimum possible value of a is

Options

A. `1/5`
B. `5/26`
C. `3/28`
D. `2/43`

Answer

To solve the problem step by step, we need to find the minimum possible integral value of \( a \) such that the graphs of the functions \( f(x) = ax^2 + 2bx + b \) and \( g(x) = 5x^2 - 3bx - a \) intersect for all real values of \( b \).

### Step 1: Set the functions equal to each other
To find the points of intersection, we set \( f(x) = g(x) \):
\[ ax^2 + 2bx + b = 5x^2 - 3bx - a \]

### Step 2: Rearrange the equation
Rearranging gives us:
\[ ax^2 - 5x^2 + 2bx + 3bx + b + a = 0 \]
This simplifies to:
\[ (a - 5)x^2 + (5b)x + (b + a) = 0 \]

### Step 3: Identify the coefficients
From the quadratic equation \( Ax^2 + Bx + C = 0 \), we identify:
- \( A = a - 5 \)
- \( B = 5b \)
- \( C = b + a \)

### Step 4: Apply the discriminant condition
For the quadratic to have real roots, the discriminant must be non-negative:
\[ B^2 - 4AC \geq 0 \]
Substituting our coefficients:
\[ (5b)^2 - 4(a - 5)(b + a) \geq 0 \]
This simplifies to:
\[ 25b^2 - 4(a - 5)(b + a) \geq 0 \]

### Step 5: Expand the discriminant expression
Expanding the expression gives:
\[ 25b^2 - 4[(a - 5)b + (a - 5)a] \geq 0 \]
This leads to:
\[ 25b^2 - 4(a - 5)b - 4(a^2 - 5a) \geq 0 \]

### Step 6: Rearranging into standard form
Rearranging gives:
\[ 25b^2 - 4(a - 5)b + (20a - 4a^2) \geq 0 \]
This is a quadratic in \( b \) with:
- Coefficient of \( b^2 = 25 \)
- Coefficient of \( b = -4(a - 5) \)
- Constant term = \( 20a - 4a^2 \)

### Step 7: Ensure the quadratic in \( b \) is non-negative
For this quadratic to be non-negative for all \( b \), we need:
1. The discriminant must be non-positive:
\[ (-4(a - 5))^2 - 4 \cdot 25 \cdot (20a - 4a^2) \leq 0 \]

### Step 8: Calculate the discriminant
Calculating the discriminant:
\[ 16(a - 5)^2 - 100(20a - 4a^2) \leq 0 \]

### Step 9: Simplifying the inequality
Expanding and simplifying:
\[ 16(a^2 - 10a + 25) - 2000a + 400a^2 \leq 0 \]
Combining like terms gives:
\[ 416a^2 - 2000a + 400 \leq 0 \]

### Step 10: Solve the quadratic inequality
To find the roots of \( 416a^2 - 2000a + 400 = 0 \):
Using the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{2000 \pm \sqrt{(-2000)^2 - 4 \cdot 416 \cdot 400}}{2 \cdot 416} \]

### Step 11: Calculate the roots
Calculating the discriminant:
\[ 4000000 - 665600 = 3334400 \]
Thus:
\[ a = \frac{2000 \pm \sqrt{3334400}}{832} \]

### Step 12: Find the minimum integral value of \( a \)
After calculating the roots, we find the range of \( a \) and identify the integral values. The minimum integral value of \( a \) that satisfies the inequality is determined to be \( 1 \).

### Final Answer
The minimum possible integral value of \( a \) is:
\[ \boxed{1} \]