Q118 / 160 · WBJEE 2015 · Mathematics
If the straight line $(a-1) x-b y+4=0$ is normal to the hyperbola $x y=1$, then which of the following does not hold?
- Correct answerA$a>1, b>0$
- B$a>1, b < 0$
- Correct answerC$a < 1, b < 0$
- D$a < 1, b>0$
Solution
Given equation of straight line is
$$
(a-1) x-b y+4=0
$$
Slope $=-\frac{\text { Coefficient of } x}{\text { Coefficient of } y}$ $=\frac{-(a-1)}{-b}=\frac{a-1}{b}$
Equation of hyperbola is $x y=1$
$\Rightarrow \quad y=\frac{1}{x}$
$\Rightarrow \quad \frac{d y}{d x}=-\frac{1}{x^{2}}$
Slope of normal $=x^{2}>0$
$\therefore \quad \frac{a-1}{b}>0$
$\Rightarrow \quad a-1>0 \quad$ and $\quad b>0$
or $\quad a-1 < 0$ and $b < 0$ $\Rightarrow \quad a>1, b>0$ or $a < 1, b < 0$
Hence, the correct answer is options: 'A', 'C'.