Question

The values of parameter such that the line $\left(\log _2\left(1+5 a-a^2\right)\right) x-5 y-\left(a^2-5\right)=0$ is a normal to the curve $x y=1$, may lie in the interval

• A. $(-\infty, 0)$ $
• B. (5, \infty)$
• C. $(0,5)$
• D. $(-\infty, 0) \cup(5, \infty)$

Answer 1

Answer: (C)

$x y=1 \Rightarrow y=\frac{1}{x} \Rightarrow \frac{d y}{d x}=-\frac{1}{x^2}$
Slope of normal $=x^2>0,(x \neq 0)$
Slope of given line $\frac{\log _2\left(1+5 a-a^2\right)}{5}>0$
$1+5 a-a^2 >1 $
$a^2-5 a< 0 $
$a \in(0,5) .$