Question
If $$x = \alpha ,\beta $$ satisfies both the equations $${\cos ^2}x + a\cos x + b = 0$$ and $${\sin ^2}x + p\sin x + q = 0$$ then the relation between $$a, b, p$$ and $$q$$ is
A.
$$1 + b + {a^2} = {p^2} - q - 1$$
B.
$${a^2} + {b^2} = {p^2} + {q^2}$$
C.
$$2\left( {b + q} \right) = {a^2} + {p^2} - 2$$
D.
None of these
Answer :
$$2\left( {b + q} \right) = {a^2} + {p^2} - 2$$
Solution :
$$\eqalign{
& \cos \alpha + \cos \beta = - a,\cos \alpha \cdot \cos \beta = b\,\,{\text{and }}\sin \alpha + \sin \beta = - p,\sin \alpha \cdot \sin \beta = q. \cr
& \therefore \,\,{a^2} + {p^2} = 2 + 2\cos \left( {\alpha - \beta } \right)\,\,{\text{and }}b + q = \cos \left( {\alpha - \beta } \right). \cr
& \therefore \,\,{a^2} + {p^2} = 2 + 2\left( {b + q} \right). \cr} $$