Question

If $x=\alpha ,\beta$ satisfies both the equation ${\cos }^{2}x+a\cos x+b=0$ and${\sin }^{2}x+p\sin x+q=0$ then 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. $2(b+q)=a^2+p^2-2$
• D. none of these

Answer 1

Answer: (C)

$2(b+q)=a^2+p^2-2$

${\cos }^{2}x+a\cos x+b=0$
$\Rightarrow \cos \alpha +\cos \beta =-a⟶\left(1\right),\cos \alpha .\cos \beta =b$
and ${\sin }^{2}x+p\sin x+q=0$
$\Rightarrow \sin \alpha +\sin \beta =-p⟶\left(2\right),\sin \alpha .\sin \beta =q$
squaring and adding (1) and (2),
$2+2(\cos \alpha .\cos \beta +\sin \alpha .\sin \beta )=a^2+p^2$
$\Rightarrow 2(b+q)=a^2+p^2-2$