Question

If $b>a$, then the equation $(x-a)(x-b)$ $-1=0$ has

• A. both roots in $[a,b]$
• B. both roots in $(-\infty ,a)$
• C. both roots in $(b,\infty )$
• D. one root in $(-\infty ,a)$ and other in $(b,\infty )$.

Answer 1

Answer: (D)

Graph of $y=(x-a)(x-b)-1$ is given in Fig. 3.17.
It is a parabola which open upwards. Also, $y<0$ for $x=a$ and $x=b$.
$\therefore y=(x-a)(x-b)-1$ meets the $x$-axis at two points once in $(-\infty ,a)$ and once in $(b,\infty )$. Thus, one root lies in $(-\infty ,a)$ and one in $(b,\infty )$