If `a lt c lt b` then the roots of the equation `(a−b)x^2 +2(a+b−2c)x+1=0` are

Options

A. imaginary
B. real
C. one real and imaginary
D. equal and imaginary

Answer

To solve the problem, we need to analyze the quadratic equation given by:

\[ (a - b)x^2 + 2(a + b - 2c)x + 1 = 0 \]

We need to determine the nature of the roots based on the condition \(a < c < b\).

### Step 1: Identify coefficients
In the quadratic equation of the form \(Ax^2 + Bx + C = 0\):
- \(A = a - b\)
- \(B = 2(a + b - 2c)\)
- \(C = 1\)

### Step 2: Calculate the discriminant
The discriminant \(D\) of a quadratic equation is given by:

\[ D = B^2 - 4AC \]

Substituting the values of \(A\), \(B\), and \(C\):

\[ D = [2(a + b - 2c)]^2 - 4(a - b)(1) \]

### Step 3: Simplify the discriminant
Calculating \(D\):

\[ D = 4(a + b - 2c)^2 - 4(a - b) \]

Factoring out the common term:

\[ D = 4\left[(a + b - 2c)^2 - (a - b)\right] \]

### Step 4: Further simplify the expression
Now we need to expand and simplify the expression inside the brackets:

\[ (a + b - 2c)^2 = a^2 + b^2 + 4c^2 + 2ab - 4ac - 4bc \]

Thus,

\[ D = 4\left[a^2 + b^2 + 4c^2 + 2ab - 4ac - 4bc - (a - b)\right] \]

### Step 5: Analyze the sign of the discriminant
Given \(a < c < b\):
- \(a - c < 0\) (meaning \(a - c\) is negative)
- \(b - c > 0\) (meaning \(b - c\) is positive)

This implies that \(a - b < 0\) (since \(a < b\)) and thus \(a - b\) is negative.

### Step 6: Conclusion about the roots
Since we have established that \(D < 0\) (the discriminant is negative), it indicates that the roots of the quadratic equation are imaginary.

### Final Answer
The roots of the equation \((a - b)x^2 + 2(a + b - 2c)x + 1 = 0\) are imaginary.

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