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-b)/(2)gt a`
D. `(c+2b)/(8)gt a`

Answer

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

### 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
\[ b + c - 2a > 0 \]

### Step 4: Combine the conditions
From the discriminant condition \( b^2 + 8ac < 0 \) and the rearranged condition \( b + c - 2a > 0 \), we can analyze the implications of these inequalities.

### Step 5: Evaluate \( f(1) \)
Let’s evaluate \( f(1) \):
\[ f(1) = c(1)^2 + b(1) - 2a = c + b - 2a \]
From the condition \( a < \frac{b+c}{2} \), we have:
\[ c + b - 2a > 0 \]
This implies:
\[ f(1) > 0 \]

### Step 6: Evaluate \( f(0) \)
Now, evaluate \( f(0) \):
\[ f(0) = c(0)^2 + b(0) - 2a = -2a \]
Since \( a < 0 \), it follows that:
\[ -2a > 0 \]
Thus, \( f(0) > 0 \).

### Step 7: Conclusion about the coefficients
Since \( f(x) \) is positive at both \( x = 0 \) and \( x = 1 \) and the discriminant is negative, we can conclude that the quadratic opens upwards (since \( c > 0 \)) and remains positive for all \( x \in \mathbb{R} \).

### Step 8: Coefficient conditions
From the analysis, we can conclude:
1. \( a < 0 \)
2. \( c > 0 \)

### Final Answer
Based on the analysis, we can deduce that the conditions imply certain relationships among \( a \), \( b \), and \( c \). The options provided in the original question would need to be evaluated based on these conclusions.