If the quadratic equation `ax^2+bx+c=0` has equal roots where a, b, c denotes the lengths of the sides opposite to vertex A, B and C of the `DeltaABC` respectively then find the number of integers in the range of `sinA/sinC+sinC/sinA`

Answer

To solve the problem, we need to find the number of integers in the range of the expression \( \frac{\sin A}{\sin C} + \frac{\sin C}{\sin A} \) given that the quadratic equation \( ax^2 + bx + c = 0 \) has equal roots, where \( a, b, c \) are the lengths of the sides opposite to vertices \( A, B, \) and \( C \) of triangle \( ABC \).

### Step-by-Step Solution:

1. **Condition for Equal Roots**:
The quadratic equation \( ax^2 + bx + c = 0 \) has equal roots if its discriminant is zero:
\[ b^2 - 4ac = 0 \]
This implies:
\[ b^2 = 4ac \]

2. **Using the Sine Rule**:
According to the sine rule in triangle \( ABC \):
\[ a = 2R \sin A, \quad b = 2R \sin B, \quad c = 2R \sin C \]
where \( R \) is the circumradius of the triangle.

3. **Substituting Values into the Discriminant Condition**:
Substitute \( a, b, c \) into the equation \( b^2 = 4ac \):
\[ (2R \sin B)^2 = 4(2R \sin A)(2R \sin C) \]
Simplifying this gives:
\[ 4R^2 \sin^2 B = 16R^2 \sin A \sin C \]
Dividing both sides by \( 4R^2 \) (assuming \( R \neq 0 \)):
\[ \sin^2 B = 4 \sin A \sin C \]

4. **Finding the Expression**:
We need to find the range of:
\[ \frac{\sin A}{\sin C} + \frac{\sin C}{\sin A} \]
Let \( x = \frac{\sin A}{\sin C} \). Then:
\[ \frac{\sin C}{\sin A} = \frac{1}{x} \]
Thus, the expression becomes:
\[ x + \frac{1}{x} \]

5. **Finding the Minimum Value**:
The function \( f(x) = x + \frac{1}{x} \) has a minimum value of 2, which occurs when \( x = 1 \) (i.e., \( \sin A = \sin C \)). Therefore:
\[ x + \frac{1}{x} \geq 2 \]

6. **Finding the Maximum Value**:
Since \( \sin A \) and \( \sin C \) are bounded by 0 and 1, the maximum value of \( x + \frac{1}{x} \) occurs as \( x \) approaches either 0 or infinity. However, in the context of a triangle, we are limited by the triangle inequality and the sine values.

The maximum value can be approached but not reached, hence we can say:
\[ x + \frac{1}{x} \to \infty \text{ as } x \to 0 \text{ or } x \to \infty \]

7. **Conclusion**:
Thus, the range of \( \frac{\sin A}{\sin C} + \frac{\sin C}{\sin A} \) is:
\[ [2, \infty) \]
The integers in this range are \( 2, 3, 4, \ldots \).

### Final Answer:
The number of integers in the range is infinite, starting from 2.