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Question: Answered & Verified by Expert
If the line $a x+b y+c=0, a b \neq 0,$ is a tangent
to the curve $x y=1-2 x,$ then
MathematicsApplication of DerivativesWBJEEWBJEE 2017
Options:
  • A $a>0, b < 0$
  • B $a>0, b>0$
  • C $a < 0, b>0$
  • D $a < 0, b < 0$
Solution:
1353 Upvotes Verified Answer
The correct answers are: $a>0, b>0$, $a < 0, b < 0$
We have, $xy=1-2 x$
$\Rightarrow y=\frac{1-2 x}{x}$
$\Rightarrow \frac{d y}{d x}=\frac{-2 x-(1-2 x) \cdot 1}{x^{2}}$
$\quad=\frac{-2 x-1+2 x}{x^{2}}=\frac{-1}{x^{2}} < 0$
since, $a x+b y+c=0$ is tangent to the curve $xy=1-2 x$
$\therefore \quad \frac{-a}{b} < 0 \Rightarrow \frac{a}{b}>0$
$\Rightarrow$ Either $a>b>0$ or $a < 0, b < 0$

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