Question

If \( \sin \alpha \) and \( \cos \alpha \) are the roots of the equation \( ax^2 + bx + c = 0 \), then

• A. \( a^2 - b^2 + 2ac = 0 \)
• B. \( (a-c)^2 = b^2 + c^2 \)
• C. \( a^2 + b^2 - 2ac = 0 \)
• D. \( a^2 + b^2 + 2ac = 0 \)
• E. \( a + b + c = 0 \)

Hint:

Always use trigonometric identities when roots involve sine and cosine.

Answer 1

The Correct Option is D

Concept: Roots satisfy: \[ \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a} \]

Step 1: Use identity.
\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]

Step 2: Express in roots form.
\[ (\sin \alpha + \cos \alpha)^2 = 1 + 2\sin \alpha \cos \alpha \]

Step 3: Substitute using sums/products.
\[ \left(-\frac{b}{a}\right)^2 = 1 + 2\frac{c}{a} \]

Step 4: Simplify equation.
\[ \frac{b^2}{a^2} = 1 + \frac{2c}{a} \]

Step 5: Multiply by \( a^2 \).
\[ b^2 = a^2 + 2ac \Rightarrow a^2 + b^2 + 2ac = 0 \]