If the equation `cx^(2)+bx-2a=0` has no real roots and `a lt (b+c)/(2)` then

Options

A. `a c lt 0`
B. `a lt 0`
C. `(c-a)/(2)gt a`
D. `(c+2b)/(8)gt a`

Answer

To solve the problem step by step, we will analyze the given quadratic equation and the conditions provided.

### Step 1: Understand the given quadratic equation
The quadratic equation is given as:
\[ cx^2 + bx - 2a = 0 \]

### Step 2: Condition for no real roots
For the quadratic equation to have no real roots, the discriminant must be less than zero. The discriminant \( D \) for the equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
In our case, the discriminant is:
\[ D = b^2 - 4c(-2a) = b^2 + 8ac \]
For the equation to have no real roots, we need:
\[ D < 0 \]
Thus, we have:
\[ b^2 + 8ac < 0 \]

### Step 3: Analyze the condition \( a < \frac{b+c}{2} \)
We are also given the condition:
\[ a < \frac{b+c}{2} \]
This can be rearranged to:
\[ 2a < b + c \]
or
\[ b + c > 2a \]

### Step 4: Evaluate \( f(1) \)
Now, we evaluate the function at \( x = 1 \):
\[ f(1) = c(1)^2 + b(1) - 2a = c + b - 2a \]
We know that \( f(1) > 0 \) implies:
\[ c + b - 2a > 0 \]
This can be rearranged to:
\[ c + b > 2a \]

### Step 5: Combine the inequalities
From the previous steps, we have:
1. \( b^2 + 8ac < 0 \)
2. \( c + b > 2a \)

### Step 6: Evaluate \( f(0) \)
Next, we evaluate the function at \( x = 0 \):
\[ f(0) = c(0)^2 + b(0) - 2a = -2a \]
Since we know \( f(x) > 0 \) for all \( x \), we have:
\[ -2a > 0 \]
This implies:
\[ a < 0 \]

### Conclusion
Thus, we conclude that:
\[ a < 0 \]

### Final Answer
The correct option is:
- \( a < 0 \)