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

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

$\Rightarrow f\left(a\right)f\left(b\right)<0\Rightarrow onerootliebetween(a,b)$

$\Rightarrow f\left(b\right)f\left(c\right)<0\Rightarrow onerootliebetween(b,C)$

both roots lie between $(a,C)$

$\therefore$all the options are correct