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)

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

$ax+by+c=0$ ........ $\left(i\right)$

Given curve is $xy=1-2x$

Tangent is obtained by differentiating the above equation of curve

So, $y=\frac{1-2x}{x}$

$⟹y=\frac{1}{x}-2$

$⟹\frac{dy}{dx}=-\frac{1}{x^2}$

Since, $x^2>0$ for every real $x$

$\therefore -\frac{1}{x^2}<0$

$\therefore \frac{dy}{dx}<0$

From the line $\left(i\right)$, we get

$\frac{dy}{dx}=-\frac{a}{b}$

$\therefore \frac{dy}{dx}<0$ if both $a,b$ are negative or both are positive

Thus, the line in $\left(i\right)$ is tangent to the given curve if either $a<0,b<0$ or $a>0,b>0$.