Question

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

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

Answer 1

Answer: (C)

$a < 0, b < 0$

Answer 2

Slope of $ax+by+c=0$ is $-\frac{a}{b}$ Also $y=\frac{4}{x}$ $\Rightarrow \frac{dy}{dx}=-\frac{4}{x^{2}}$ $\therefore -\frac{a}{b}=-\frac{4}{x^{2}}$ $\Rightarrow \frac{a}{b}=-\frac{4}{x^{2}} > 0$ $\therefore a < 0, b < 0$