Multiple choice

If $\displaystyle \sin \alpha, : \cos \alpha$ are the roots of the equation $\displaystyle cx^{2} + bx + a = 0$, then $\displaystyle a, : b, : c$ satisfies

  1. $\displaystyle c^{2} + 2ac + b^{2} = 0$
  2. $\displaystyle b^{2} - 2ac - c^{2} = 0$
  3. $\displaystyle 2ac - b^{2} - c^{2} = 0$
  4. $\displaystyle b^{2} + 2ac - c^{2} = 0$
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B Correct answer
Explanation

Roots are sin(a) and cos(a). Sum = sin(a) + cos(a) = -b/c. Product = sin(a)cos(a) = a/c. Squaring the sum: sin^2(a) + cos^2(a) + 2sin(a)cos(a) = b^2/c^2. 1 + 2(a/c) = b^2/c^2. Multiplying by c^2: c^2 + 2ac = b^2, or b^2 - 2ac - c^2 = 0.

AI explanation

Given sin alpha and cos alpha are the roots of cx squared + bx + a = 0, their sum is -b/c and their product is a/c. Using the trigonometric identity sin squared alpha + cos squared alpha = 1, we square the sum: (sin alpha + cos alpha) squared = 1 + 2(sin alpha)(cos alpha). Substituting the root relationships gives (-b/c) squared = 1 + 2(a/c), which simplifies to b squared divided by c squared = 1 + 2a divided by c. Multiplying by c squared results in b squared = c squared + 2ac, or b squared - 2ac - c squared = 0.

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