Multiple choice

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

  5. NA
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Roots are sin(theta) and cos(theta). Sum = sin+cos = -b/a. Product = sin*cos = c/a. (sin+cos)^2 = 1 + 2sin*cos = b^2/a^2. 1 + 2(c/a) = b^2/a^2. Multiplying by a^2: a^2 + 2ac = b^2, so a^2 - b^2 + 2ac = 0.

AI explanation

Using Vieta's formulas for the equation ax2 + bx + c = 0, the sum of the roots is sinθ + cosθ = -b/a and the product is sinθ cosθ = c/a. The trigonometric identity sin2θ + cos2θ = 1 can be rewritten as (sinθ + cosθ)2 - 2 sinθ cosθ = 1. Substituting the sum and product values gives (-b/a)2 - 2(c/a) = 1, which simplifies to b2 - 2ac = a2, or a2 - b2 + 2ac = 0.

Keep practicing — related questions