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

The correct option is C $a<0,b>0$

Differentiating the equation of curve xy = 1
We have x$\frac{dy}{dx}$ + y = 0 $\Rightarrow \frac{dy}{dx}$ = -$\frac{y}{x}$
Hence the slope of normal = $\frac{x}{y}$ .
Moreover the slope of the line aX + bY + c = 0 is -$\frac{a}{b}$ So we have $\frac{x}{y}=-\frac{a}{b}$, i.e. bx + ay = 0
solving this with xy = 1, we have $x^2$ = -$\frac{a}{b}$ So we must have $\frac{a}{b}<$ 0 i.e., a $>$ 0, b $<$ 0 or
a $<$t 0, b $>$ 0.