If the line ax + by + c = 0 is a tangent to the curve xy = 9, then(a)…

We have

$x y=9 \Rightarrow y=\frac{9}{x} \Rightarrow \frac{d y}{d x}=-\frac{9}{x^2}$            ........(i)

Let $a x+b y+c=0$ be a tangent to the curve $x y=9$ at $\left(x_1, y_1\right)$

Then,

$\left(\frac{d y}{d x}\right)_{\left(x_1, y_1\right)}$ = Slope of the line ax + by + c = 0

$\Rightarrow -\frac{9}{x_1^2}=\frac{-a}{b}$

$\Rightarrow \frac{x_1^2}{9}=\frac{b}{a}$

$\Rightarrow \frac{b}{a}>0$

⇒ a and b are the same sign.

⇒  (a > 0 and b > 0) or, (a < 0 and b < 0)