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If the line $y=-4 x+b$ is tangent to the curve $y=\frac{1}{x}$, then $b$ equals
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Verified Answer
The correct answer is:
\pm 4
The slope of line $y=-4 x+b$ is $m=-4$.
Also, slope of tangent of the curve $y=\frac{1}{x}$ is given by $\frac{d y}{d x}=\frac{-1}{x^2}$.
Since, given line is tangent to the curve.
$$
\begin{array}{ll}
\therefore & \frac{-1}{x^2}=-4 \Rightarrow x^2=\frac{1}{4} \\
\Rightarrow & x= \pm \frac{1}{2} \text { or } y= \pm 2
\end{array}
$$
Put these values in $y=-4 x+b$, we get $b= \pm 4$.
Also, slope of tangent of the curve $y=\frac{1}{x}$ is given by $\frac{d y}{d x}=\frac{-1}{x^2}$.
Since, given line is tangent to the curve.
$$
\begin{array}{ll}
\therefore & \frac{-1}{x^2}=-4 \Rightarrow x^2=\frac{1}{4} \\
\Rightarrow & x= \pm \frac{1}{2} \text { or } y= \pm 2
\end{array}
$$
Put these values in $y=-4 x+b$, we get $b= \pm 4$.
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