Question

If the line $ax+by+c=0$ is a normal to the curve $xy=1$, then

• A. $a>0,b>0$
• B. $a>0,b<0$ or $a<0,b>0$
• C. $a<0,b<0$
• D. none of these

Answer 1

Answer: (B)

Let the line ax + by + c = 0 be normal to the curve xy = 1 at the point $(x^{\prime },y^{\prime })$, then $x^{\prime }{y}^{\prime }=1$ .....(1) [ pt $(x^{\prime },y^{\prime })$ lies on the curve]
Also differentiating the curve xy = 1 with respect to x we get
$y+x\frac{dy}{dx}=0\Rightarrow \frac{dy}{dx}=-\frac{y}{x}$
$\Rightarrow \frac{dy}{dx}{)}_{\left(x^{\prime },y^{\prime }\right)}=\frac{-y^{\prime }}{x^{\prime }}$
$\therefore$ Slope of normal $=\frac{x^{\prime }}{y^{\prime }}$
Also equation of normal suggests, slope of normal $=\frac{-a}{b}$
$\therefore$ We must have,
$\frac{x^{\prime }}{y^{\prime }}=\frac{a}{b}$ .......(2)
Now from eq. (1), $x^{\prime }{y}^{\prime }>0\Rightarrow x^{\prime },y^{\prime }$ are of same sign
$\Rightarrow \frac{x^{\prime }}{y^{\prime }}=+ve\Rightarrow -\frac{a}{b}=+ve\Rightarrow \frac{a}{b}=-ve$
$\Rightarrow$ a and b are of opposite sign.
$\Rightarrow$ either a < 0 and b > 0 or a > 0 and b < 0.