Question

If the straight line $(a-1)x-by+4=0$ is normal to the hyperbola $xy=1$, then which of the following 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)

$a<1,b<0$
$a>1,b>0$

Given equation of straight line is $(a-1)x-by+4=0$
Slope $=\frac{\text{Coefficient of x}}{\text{Coefficient of y}}$

$=\frac{-(a-1)}{-b}=\frac{a-1}{b}$

Equation of hyperbola is $xy=1$
$\Rightarrow y=\frac{1}{x}$

$\Rightarrow \frac{dy}{dx}=-\frac{1}{x^2}$

slope of normal $=x^2>0$

$\therefore \frac{a-1}{b}>0$

$\Rightarrow a-1>0$ and $b>0$

or $a-1<0$ and $b<0$

$\Rightarrow a>1,b>0$ or $a<1,b<0$