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

To determine the correct relationship, we start by noting that the solutions to the quadratic equation \(x^2 + ax - c = 0\) are given as \(\sin 2\theta\) and \(\cos 2\theta\). The sum and product of the roots of a quadratic equation \(ax^2 + bx + c = 0\) are given by:

• Sum of roots = \(-\frac{b}{a}\)
• Product of roots = \(\frac{c}{a}\)

In this case, the standard form of the equation is \(x^2 + ax - c = 0\), which gives us:

• Sum of roots \(= -a\)
• Product of roots \(= -c\)

Therefore, we have the following equations based on the sum and product:

1. \(\sin 2\theta + \cos 2\theta = -a\)
2. \(\sin 2\theta \cdot \cos 2\theta = -c\)

We already know the identity for \(\sin 2\theta\) and \(\cos 2\theta\):

• \(\sin^2 2\theta + \cos^2 2\theta = 1\)

Now, using the identity:

• \((\sin 2\theta + \cos 2\theta)^2 = \sin^2 2\theta + \cos^2 2\theta + 2\sin 2\theta \cdot \cos 2\theta\)
• Substitute \(\sin 2\theta + \cos 2\theta = -a\) and \(\sin 2\theta \cdot \cos 2\theta = -c\) into the identity:
• \((-a)^2 = 1 + 2(-c)\)
• \(a^2 = 1 - 2c\)

Rearranging the equation gives:

• \(a^2 + 2c - 1 = 0\)

Therefore, the correct answer is the option that matches this equation:

a² + 2c - 1 = 0