Question

If the line $ax+by+c=0$, where $a,b,c$ are non-zero real numbers, is normal to the rectangular hyperbola $xy=1$, then which of the following is/are true ?

• 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)

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

$xy=1$

Slope of tangent at $(x_1,y_1)$ is $\Rightarrow \frac{dy}{dx}{|}_{x=x_1}=\frac{-1}{x_1^2}$
Hence, the slope of normal is given by $x_1^2$, which is positive.
So, $\frac{-a}{b}>0$ or $a$ and $b$ have opposite signs.
So, options B and C are correct.