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 $( x', y' )$, then $x' y' = 1 $ .....(1) [ pt $(x', y') $ 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}\bigg)_{\left(x',y'\right)} = \frac{-y'}{x'} $
$\therefore $ Slope of normal $ = \frac{x'}{y'}$
Also equation of normal suggests, slope of normal $ = \frac{-a}{b}$
$ \therefore $ We must have,
$\frac{x'}{y'} = \frac{a}{b} $ .......(2)
Now from eq. (1), $x'y' > 0\, \Rightarrow \, x', y'$ are of same sign
$\Rightarrow \frac{x'}{y'} = +ve \Rightarrow - \frac{a}{b} = + ve \Rightarrow \frac{a}{b} = -ve$
$ \Rightarrow $ a and b are of opposite sign.
$\Rightarrow $ either a < 0 and b > 0 or a > 0 and b < 0.