Question

If sinθ and cosθ are the roots of the equation ax² + bx + c = 0, then which one of the following is correct?
1. a² + b² - 2ac = 0
2. -a2 + b2 + 2ac = 0
3. a2 - b2 + 2ac = 0
4. a2 + b2 + 2ac = 0

Answer 1

Correct Answer - Option 3 : a2 - b2 + 2ac = 0

Formula used:

For any quadratic equation ax2 + bx + c = 0,

when α and β are the roots then

α + β = \(\rm \frac{-b}{a}\)

αβ = \(\rm \frac{c}{a}\)

(sinθ + cos θ)2 = sin2θ + cos2θ + 2 sin θ cos θ

sin2θ + cos2θ = 1

Calculation:

According to the question

sin θ and cos θ are the roots of the equation ax² – bx + c = 0,

sin θ + cos θ = \(\rm \frac{b}{a}\)     ----(i)

sin θ . cos θ = \(\rm \frac{c}{a}\)     ----(ii)

(sinθ + cos θ)² = sin²θ + cos²θ + 2 sin θ cos θ    ----(iii)

Form (i), (ii) and (iii), we get 

\(\rm \frac{b^{2}}{a^{2}} = 1 + \frac{2c}{a}\)

⇒ b² = a² + 2ac

⇒ a² - b² + 2ac = 0

∴ The correct relation between a, b and c is a2 - b2 + 2ac = 0.