Question

If the line $ax+by+c=0,ab\ne 0$, is a tangent to the curve $xy=1-2x$, then

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

Answer 1

Answer: (B), (D)

The correct options are
B $a>0,b>0$
D $a<0,b<0$

$xy=1-2x$
$\Rightarrow y=\frac{1}{x}-2$
$\Rightarrow \frac{dy}{dx}=\frac{-1}{x^2}<0$
This means slope of tangent at any point is negative.

$ax+by+c=0$ is a tangent to the given curve.
Its slope is $\frac{-a}{b}$.
$\therefore \frac{-a}{b}<0$
$\Rightarrow \frac{a}{b}>0$
$\Rightarrow a>0,b>0$ or $a<0,b<0$