Question

If sin 2θ and cos 2θ are solutions of x² + ax - c = 0, then

• A. a² - 2c - 1 = 0
• B. a² + 2c - 1 = 0
• C. a² + 2c + 1 = 0
• D. a² - 2c + 1 = 0

Answer 1

The Correct Option is B

Given that $\sin 2\theta$ and $\cos 2\theta$ are solutions to the quadratic equation $x^2 + ax - c = 0$, we need to establish a relationship between $a$ and $c$.

1. Using Vieta's Formulas:
For the quadratic equation $x^2 + ax - c = 0$ with roots $\sin 2\theta$ and $\cos 2\theta$, we have:
Sum of roots: $\sin 2\theta + \cos 2\theta = -a$
Product of roots: $\sin 2\theta \cos 2\theta = -c$

2. Squaring the Sum of Roots:
$(\sin 2\theta + \cos 2\theta)^2 = (-a)^2$
$\sin^2 2\theta + 2\sin 2\theta \cos 2\theta + \cos^2 2\theta = a^2$

3. Simplifying Using Trigonometric Identities:
Using the Pythagorean identity $\sin^2 2\theta + \cos^2 2\theta = 1$:
$1 + 2\sin 2\theta \cos 2\theta = a^2$

4. Substituting the Product of Roots:
From Vieta's formula, we know $\sin 2\theta \cos 2\theta = -c$, so:
$1 + 2(-c) = a^2$
$1 - 2c = a^2$

5. Final Relationship:
Rearranging the equation gives the required relationship:
$a^2 + 2c = 1$

Final Answer:
The relationship between $a$ and $c$ is ${a^2 + 2c = 1}$.