Question

The equation of the line parallel to $x$ axis and tangent to the curve $y=\frac{1}{x^2+2x+5}$ is

• A. $y=\frac{1}{4}$
• B. $y=4$
• C. $y=\frac{1}{2}$
• D. $y=2$

Answer 1

The correct option is A

$y=\frac{1}{4}$

Explanation of correct answer:

Tangent on the curve:

Given: $y=\frac{1}{x^2+2x+5}$………$\left(1\right)$

Let the equation of line which is parallel to $x$ axis is , $y=c$………$\left(2\right)$

The line is a tangent to the curve.

Therefore, $\mathrm{Slope}\mathrm{of}\mathrm{the}\mathrm{curve}=\mathrm{Slope}\mathrm{of}\mathrm{the}\mathrm{line}$

$\Rightarrow$ $\frac{d}{dx}\left(\frac{1}{x^2+2x+5}\right)=\frac{d}{dx}c$

$\Rightarrow$ $\frac{-(2x+2)}{{(x^2+2x+5)}^2}=0$

$\Rightarrow$ $2x+2=0$

$\Rightarrow$ $x=-1$

From equation$\left(1\right)$,

$$\begin{gathered} y=\frac{1}{1-2+5} \\ \Rightarrow y=\frac{1}{4} \end{gathered}$$

From equation$\left(2\right)$,

$c=\frac{1}{4}$

Hence, the required equation of the line is $y=\frac{1}{4}$.

Hence,option $A$ is the correct answer.