Question

If the straight line $(a-1)x-by+4=0$ is normal to the hyperbola $xy=1$ then which of the followings does not hold?

• A. $a>1,b>0$
• B. $a>1,b<0$
• C. $a<1,b<0$
• D. $a<1,b>0$

Answer 1

Answer: (A), (C)

The correct options are
A $a>1,b>0$
C $a<1,b<0$

$xy=1$
Differentiating both sides w.r.t. $x$, we get
$x\frac{dy}{dx}+y=0$
$\Rightarrow \frac{dy}{dx}=-\frac{y}{x}$
Therefore, slope of tangent $=-\frac{y}{x}$

Given the equation of normal :
$by=(a-1)x+4$
$\Rightarrow y=\frac{(a-1)}{b}x+\frac{4}{b}$
Slope of normal $=\frac{a-1}{b}$

So, $\left(\frac{-y}{x}\right)\left(\frac{a-1}{b}\right)=-1<0$
As $xy=1$
Thus, $\frac{y}{x}$ cannot be negative.
Thus, $a-1<0,b<0$
$\Rightarrow a<1,b<0$
or $a-1>0,b>0$
$\Rightarrow a>1,b>0$