Question

Let , $f\left(x\right)=a{x}^2+bx+c,g\left(x\right)=a{x}^2+px+qwherea,b,c,q,p,ϵRandb\ne p$. If their discriminants are equal and f(x) = g(x) has a root , $\alpha$ then

• A. will be A.M. of the roots of f(x) = 0, g(x) = 0
• B. will be G.M of all the roots of f(x) = 0, g(x) = 0
• C. will be A.M of the roots of f(x) = 0 or g(x) = 0
• D. will be G.M of the roots of f(x) = 0 or g(x) = 0

Answer 1

The correct option is A

will be A.M. of the roots of f(x) = 0, g(x) = 0

$a{\alpha }^2+b\alpha +c=a{\alpha }^2+p\alpha +q\Rightarrow \alpha =\frac{q-c}{b-p}\to \left(i\right)And{b}^2-4ac=p^2-4aq\Rightarrow b^2-p^2=4a(c-q)\Rightarrow b+p=\frac{4a(c-q)}{b-p}=-4a\alpha \left(from\right(i\left)\right)$

$\alpha =\frac{-(b+p)}{4a}=\frac{\frac{-b}{a}-\frac{p}{a}}{4}$ which is A.M of all the roots of f(x) = 0 and g(x) = 0