Question

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

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

Answer 1

Answer: (B), (C)

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

$ax+by+c=0$
Hence
$by=-ax-c$
Or
$y=\frac{-a}{b}x-c$
Hence
$m=\frac{-a}{b}$
= Slope of normal.
Now
$x{y}^{\prime }+y=0$
Or
$y^{\prime }=\frac{-y}{x}$
Or
$y^{\prime }=\frac{-1}{x^2}$
Thus
$\frac{-1}{y^{\prime }}$
$=x^2$
Hence
$x^2=\frac{-a}{b}$
Or
Now $x^2>0$ thus
$\frac{-a}{b}$ has to be positive.
Hence we are left with 2 options.
Either
$a<0$ and $b>0$
Or
$a>0$ and $b<0$.