If tan^2 theta, sec^2 theta are the roots of ax^2 + bx + c = 0 then b^2 - a^2 = A 4ac B a^2 C 4bc D 4ab

To solve for the value of $b^2 - a^2$ given that $\tan^2 \theta$ and $\sec^2 \theta$ are the roots of the quadratic equation $ax^2 + bx + c = 0$, follow these steps:

1. Sum and Product of Roots:

   • Recognize that for a quadratic equation $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$, the sum of the roots is given by: $$\alpha + \beta = -\frac{b}{a}$$ and the product of the roots is: $$\alpha \beta = \frac{c}{a}$$
   • Here, let $\alpha = \sec^2 \theta$ and $\beta = \tan^2 \theta$.

2. Key Identity:

   • Use the trigonometric identity: $$\sec^2 \theta - \tan^2 \theta = 1$$

3. Equation Setup:

   • Using the identities for roots, we substitute: $$\alpha + \beta = \sec^2 \theta + \tan^2 \theta = -\frac{b}{a}$$ $$\alpha \beta = \sec^2 \theta \cdot \tan^2 \theta = \frac{c}{a}$$

4. Relation Application:

   • Using the relation $(\alpha + \beta)^2 = (\alpha - \beta)^2 + 4\alpha \beta$, substitute the known values: $$\left( -\frac{b}{a} \right)^2 = (\sec^2 \theta - \tan^2 \theta)^2 + 4 \frac{c}{a}$$
   • As $\sec^2 \theta - \tan^2 \theta = 1$: $$\left( -\frac{b}{a} \right)^2 = 1^2 + 4 \frac{c}{a}$$

5. Simplify:

   • Simplify the above equation: $$\frac{b^2}{a^2} = 1 + \frac{4c}{a}$$
   • Cross-multiply to clear the fraction: $$b^2 = a^2 + 4ac$$

6. Final Result:

   • Rearrange to find $b^2 - a^2$: $$b^2 - a^2 = 4ac$$

Thus, the value of $b^2 - a^2$ is 4ac.

Answer: A