If `sectheta` and `tantheta` are roots of equation `ax^(2) + bx + c=0 (a,b ne 0)` then the value of `sectheta - tan theta` is:

Options

A. `-a/b`
B. `sqrt(b^(2) - 4ac)/a`
C. `1-a/b`
D. `1 + a^(2)/b^(2)`

Answer

To solve the problem step by step, we need to determine the value of \( \sec \theta - \tan \theta \) given that \( \sec \theta \) and \( \tan \theta \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \).

### Step 1: Identify the roots
Let the roots of the equation be:
- \( r_1 = \sec \theta \)
- \( r_2 = \tan \theta \)

### Step 2: Use Vieta's formulas
According to Vieta's formulas, for a quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \)
- The product of the roots \( r_1 \cdot r_2 = \frac{c}{a} \)

Thus, we have:
1. \( \sec \theta + \tan \theta = -\frac{b}{a} \) (1)
2. \( \sec \theta \cdot \tan \theta = \frac{c}{a} \) (2)

### Step 3: Express \( \sec \theta - \tan \theta \)
We want to find \( \sec \theta - \tan \theta \). We can use the identity:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
This can be factored as:
\[ (\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1 \]

### Step 4: Substitute the sum of the roots
From equation (1), we can substitute \( \sec \theta + \tan \theta \):
\[ (\sec \theta - \tan \theta)\left(-\frac{b}{a}\right) = 1 \]

### Step 5: Solve for \( \sec \theta - \tan \theta \)
Rearranging gives:
\[ \sec \theta - \tan \theta = -\frac{a}{b} \]

### Conclusion
Thus, the value of \( \sec \theta - \tan \theta \) is:
\[ \sec \theta - \tan \theta = -\frac{a}{b} \]