Question

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

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

Answer 1

The correct option is C $(0,5)$

For $xy=1\Rightarrow \frac{dy}{dx}=-\frac{1}{x^2}$
Slope of normal to $xy=1$ for $x^2$ is > 0 $(x\ne 0)$
Slope of line (given) $=\frac{lo{g}_2(1+5a-a^2)}{5}>0$
So $1+5a-a^2>1$
$a^2-5a<0$ hence $aϵ(0,5)$