Question

If the quadratic equation $4 x^{2}-2(a+c-1) x+a c-b=0(a>b>c)$

• A. Both roots are greater than a
• B. Both roots are less than c
• C. Both roots lie between $c / 2$ and $a / 2$
• D. Both one of the roots lies between $c / 2$ and $a / 2$.

Answer 1

Answer: (D)

Here, $f(x)=(2 x-a)(2 x-c)+(2 x-b)$. So,
$f \left(\frac{ a }{2}\right)= a - b , f \left(\frac{ c }{2}\right)= c - b$
Now,
$f \left(\frac{ a }{2}\right) f \left(\frac{ c }{2}\right)=( a - b )( c - b )<0( a > b > c )$
Hence, exactly one of the roots lies between $c / 2$ and $a / 2$.