Question

If \( \sin \theta \) and \( \cos \theta \) are the roots of the equation \( ax^{2} - bx + c = 0 \), then:

• A. $a^{2} - b^{2} + 2ac = 0$
• B. $a^{2} + b^{2} - 2ac = 0$
• C. $a^{2} - b^{2} - 2ac = 0$
• D. $a^{2} + b^{2} + 2ac = 0$

Hint:

The identity $\sin^{2} \theta + \cos^{2} \theta = 1$ is the bridge between the sum and product of these roots.

Answer 1

The Correct Option is A

Step 1: Concept
Use relations between roots and coefficients: Sum $= b/a$, Product $= c/a$.
Step 2: Analysis
$\sin \theta + \cos \theta = b/a$ and $\sin \theta \cos \theta = c/a$.
Squaring the sum: $(\sin \theta + \cos \theta)^{2} = (b/a)^{2}$.
$1 + 2 \sin \theta \cos \theta = b^{2}/a^{2}$.
Step 3: Conclusion
$1 + 2(c/a) = b^{2}/a^{2} \Rightarrow a^{2} + 2ac = b^{2} \Rightarrow a^{2} - b^{2} + 2ac = 0$.
Final Answer: (A)