Question

If the graph of $f(x)=\frac{3}{x}$ and $g(x)=ax+1$ intersect at two distinct points then the range of a is equal to

• A. $(0,\infty )$
• B. $\left(-\infty ,\frac{-1}{12}\right)\cup (0,\infty )$
• C. $\left(\frac{-1}{12},0\right)\cup (0,\infty )$
• D. $\left(-\infty ,\frac{-1}{12}\right)\cup \left(\frac{-1}{12},\infty \right)$

Answer 1

Answer: (C)

Case-I: If slope of line $=a>0$ then two intersection point,
So, $a\in (0,\infty )$....(1)
Case-II: If $a<0$
${\frac{dy}{dx}|}_{P\left(a,\frac{3}{a}\right)}=\frac{-3}{a^2}=\frac{\frac{3}{a}-1}{a-0}\Rightarrow \frac{-3}{a^2}=\frac{3-a}{a^2}\Rightarrow a=6$
$\therefore \text{ Slope at }P=\frac{-1}{12}\text{ (For tangency) }$
$\therefore$ Slope at $P=\frac{-1}{12}$ (For tangency)
So, $a\in \left(\frac{-1}{12},0\right)$....(2)
Hence, $(1)\cup (2)\Rightarrow \left(\frac{-1}{12},0\right)\cup (0,\infty )$.