If the line $ax+by+c=0$ is a tangent to the curve $xy=4$, then:

Slope of the line $ax+by+c=0$ is $m_1=-\frac{a}{b}$.Curve: $y=\frac{4}{x}$.Slope of the tangent at any point $(x,y)$ is:
$$\frac{dy}{dx}=-\frac{4}{x^2}$$
Since the line is a tangent:
$$-\frac{a}{b}=-\frac{4}{x^2}⟹\frac{a}{b}=\frac{4}{x^2}$$
Since $x^2$ is always positive for any real $x\ne 0$, the ratio $\frac{a}{b}$ must be positive.$\frac{a}{b}>0$ implies that $a$ and $b$ must have the same sign (both positive or both negative).The option $a<0,b<0$ satisfies this condition.