Question

The quadratic equation, $a(x-b)(x-c)+b(x-c)(x-a)+c(x-a)(x-b)=0$, where $0>a>b>c,$ has

• A. exactly one root lying between $a\&b$
• B. exactly one root lying between $b\&c$
• C. both roots lie between $a\&c$
• D. all of these

Answer 1

The correct option is D all of these

If $f\left(p\right)f\left(q\right)<0$ then we can say that one root lie between $(p,q)$

Now $f\left(a\right)=a(a-b)(a-c)<0$

$f\left(b\right)=b(b-c)(b-a)>0$

And $f\left(c\right)=c(c-a)(c-b)<0$

$\Rightarrow f\left(a\right)f\left(b\right)<0\Rightarrow$ one root lie between$(a,b)$

$\Rightarrow f\left(b\right)f\left(c\right)<0\Rightarrow$ one root lie between $(b,c)$

both roots lie between $(a,c)$

$\therefore$all the options are correct