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

A
$a > 0, b > 0$ ,/ $a < 0, b < 0$
B
$a > 0, b < 0$
C
$a < 0, b > 0$
D
$a < 0, b < 0$

Solution:

We have,

$x y = 9 \Rightarrow y = \frac{9}{x} \Rightarrow \frac{d y}{d x} = - \frac{9}{x^{2}}$

Let $a x + b y + c = 0$ be a tangent to the curve $x y = 9$ at $(x_{1}, y_{1})$ .Then

${(\frac{d y}{d x})}_{(x_{1}, y_{1})} =$ Slope of the line $a x + b y + c = 0$

$\begin{aligned} \Rightarrow - \frac{9}{{x_{1}}^{2}} = \frac{- a}{b} \\ \Rightarrow \frac{{x_{1}}^{2}}{9} = \frac{b}{a} \\ \Rightarrow b a > 0 \end{aligned}$

=> a and b are the same sign.
$(a > 0 a n d b > 0) o r, (a < 0 a n d b < 0)$

If the line $a x + b y + c = 0$ is a tangent to the curve $x y = 9,$ then

Solution:

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