If `alpha, beta` are the roots of equation `(k+1)x^(2)-(20k+14)x+91k+40=0,(alpha lt beta),k gt 0`, then the answer the following questions
The nature of the roots of this equation is

Options

A. imaginary
B. real and distinct
C. one real only
D. none of these

Answer

To determine the nature of the roots of the quadratic equation \((k+1)x^2 - (20k + 14)x + (91k + 40) = 0\), we will use the discriminant method. The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula:

\[ D = b^2 - 4ac \]

### Step 1: Identify coefficients
From the given equation, we can identify the coefficients:
- \(a = k + 1\)
- \(b = -(20k + 14)\)
- \(c = 91k + 40\)

### Step 2: Calculate the discriminant
Now, we will substitute these coefficients into the discriminant formula:

\[ D = [-(20k + 14)]^2 - 4(k + 1)(91k + 40) \]

Calculating \(b^2\):

\[ D = (20k + 14)^2 - 4(k + 1)(91k + 40) \]

Expanding \(b^2\):

\[ D = (400k^2 + 560k + 196) - 4(k + 1)(91k + 40) \]

Now, we will calculate \(4(k + 1)(91k + 40)\):

\[ 4(k + 1)(91k + 40) = 4[91k^2 + 40k + 91k + 40] = 4[91k^2 + 80k + 40] = 364k^2 + 320k + 160 \]

Now substituting back into the discriminant:

\[ D = 400k^2 + 560k + 196 - (364k^2 + 320k + 160) \]

Combining like terms:

\[ D = (400k^2 - 364k^2) + (560k - 320k) + (196 - 160) \]
\[ D = 36k^2 + 240k + 36 \]

### Step 3: Factor the discriminant
We can factor out 36:

\[ D = 36(k^2 + 6.67k + 1) \]

### Step 4: Analyze the discriminant
Since \(k > 0\), the term \(36(k^2 + 6.67k + 1)\) is always positive because \(k^2 + 6.67k + 1\) is a quadratic expression that opens upwards and has a positive leading coefficient. Therefore, \(D > 0\).

### Conclusion
Since the discriminant \(D\) is positive, the roots of the equation are real and distinct.

### Final Answer
The nature of the roots of the equation is **Real and Distinct**.