Question

If roots of the quadratic equation $b{x}^2-2ax+a=0$ are real and distinct, where $a,b\in R$ and $b\ne 0$, then

• A. atleast one root lies in the interval $(0,1)$.
• B. no root lies in the interval $(0,1)$.
• C. atleast one root lies in the interval $(-1,0)$.
• D. none of the above.

Answer 1

Answer: (A)

Given, $b^2-2ax+a=0$
As, roots are real and distinct, so $D>0\Rightarrow 4a^2-4ab>0\Rightarrow a(a-b)>0$ ....(i)
Let $f(x)=b{x}^2-2ax+a$
So, $f(0)=a$ and $f(1)=(b-a)$
$\Rightarrow f(0)f(1)=a(b-a)=-a(a-b)$
$\Rightarrow f(0)f(1)<0$ (Using equation (i))
So, $f(x)=0$ will have a root in $(0,1)$.