Question

If sin θ and cos θ are the roots of the equation ax² – bx + c = 0, then a, b and c satisfy the relation
(a) a² + b² + 2ac = 0
(b) a² – b² + 2ac = 0
(c) a² + c² + 2ab = 0
(d) a² – b² – 2ac = 0

Answer 1

Given sinθ and cosθ are roots of ax² – bx + c = 0
Sum of roots is $\frac{b}{a}$ and product of root is $\frac{c}{a}$
i.e $\cos \theta +\sin \theta =\frac{b}{a}\mathrm{and}\cos \theta \sin \theta =\frac{c}{a}$
Since sin²θ + cos²θ = 1
$$\begin{gathered} \mathrm{i}.\mathrm{e}{\left(\sin \theta +\cos \theta \right)}^2={\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta \\ \mathrm{i}.\mathrm{e}{\left(\frac{b}{a}\right)}^2=1+2\frac{c}{a} \\ \mathrm{i}.\mathrm{e}{b}^2=a^2+2ac \\ \mathrm{i}.\mathrm{e}{a}^2-b^2+2ac \end{gathered}$$

Hence, the correct answer is option B.