Question

Compute set of values of $a$ such that $y=\frac{x^2-x}{1-ax}(xϵR$) attains all real values, is

• A. $[1,4]$
• B. $(0,4]$
• C. $(0,1]$
• D. $[1,\infty )$

Answer 1

The correct option is D $[1,\infty )$

$\begin{array}{cc}& y=\frac{x^2-x}{1-ax}\\ \Rightarrow & y-axy=x^2-x\\ \Rightarrow & x^2+x(ay-1)-y=0\end{array}$

Then, $x=\frac{(1-ay)\pm \sqrt{(ay-1{)}^2+4y}}{2}$

for all real value of $x,y$.

$(ay-1{)}^2+4y⩾0$.

$\Rightarrow a^2{y}^2+4y-2ay+1⩾0$

for above equation,

$\begin{array}{c}(4-2a{)}^2-4a^2⩽0\\ 16-16a\le 0\end{array}($ for real $y)$

$\Rightarrow a⩾1$

$\Rightarrow a\in [1,\infty )$

$\Rightarrow$ option $D$ is correct