If `ax^2+bx+c=0 and cx^2 + bx+a=0(a,b,c in R)` have a common non-real roots, then which of the following is not true ?

Options

A. `-2|a|lt |b| lt |a|`
B. `-2|c|lt b lt 2|c|`
C. `a=c`
D. None of these

Explanation

To solve the problem, we need to analyze the two quadratic equations given:

1. \( ax^2 + bx + c = 0 \)
2. \( cx^2 + bx + a = 0 \)

We are told that these two equations have a common non-real root. Let's denote the common non-real root as \( p + iq \) (where \( i \) is the imaginary unit and \( q \neq 0 \)). The other root of each equation will be \( p - iq \).

### Step 1: Sum of Roots
For the first equation \( ax^2 + bx + c = 0 \):
- The sum of the roots is given by:
\[ (p + iq) + (p - iq) = 2p = -\frac{b}{a} \]

For the second equation \( cx^2 + bx + a = 0 \):
- The sum of the roots is:
\[ (p + iq) + (p - iq) = 2p = -\frac{b}{c} \]

### Step 2: Equating the Sums
Since both equations have the same sum of roots:
\[ -\frac{b}{a} = -\frac{b}{c} \]
This implies:
\[ \frac{b}{a} = \frac{b}{c} \]
Assuming \( b \neq 0 \) (if \( b = 0 \), we would have a trivial case), we can cross-multiply:
\[ bc = ab \]
This leads us to conclude:
\[ a = c \]

### Step 3: Discriminant Condition
Since the roots are non-real, the discriminants of both equations must be less than zero.

For the first equation:
\[ D_1 = b^2 - 4ac < 0 \]
Substituting \( c = a \):
\[ D_1 = b^2 - 4a^2 < 0 \implies b^2 < 4a^2 \]

For the second equation:
\[ D_2 = b^2 - 4ca < 0 \]
Again substituting \( c = a \):
\[ D_2 = b^2 - 4a^2 < 0 \implies b^2 < 4a^2 \]

### Step 4: Conclusion on \( b \)
From the inequality \( b^2 < 4a^2 \), we can derive:
\[ |b| < 2|a| \]

### Step 5: Analyzing Options
Now, we need to check which of the given options is not true based on the derived inequalities.

1. \( |b| < 2|c| \) (True, since \( c = a \))
2. \( |b| > -2|c| \) (True, since \( |b| \) is always non-negative)
3. \( |b| < 2|a| \) (True)
4. \( |b| > -2|a| \) (True)

Thus, the statement that is not true based on our analysis would be the one that contradicts \( |b| < 2|a| \).

### Final Answer
The option that is not true is the one that suggests a different relationship for \( b \) compared to \( a \) and \( c \).

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