Question

The line, which is parallel to $x-$axis and crosses the curve $y=\sqrt{x}$ at an angle $45°$, is

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

Answer 1

The correct option is B

$y=\frac{1}{2}$

Explanation for correct option:

Given: line is parallel to the $x-$axis and crosses $y=\sqrt{x}$at an angle $45°$

Let the line parallel to the $x-$axis be $y=c$

Since the line intersects $y=\sqrt{x}$at $45°$so

The slope of the line$=\tan (45°)=1$

The slope of the curve $y=\sqrt{x}$ is $\frac{dy}{dx}$

Since the slope of the line $=\tan (45°)=\frac{dy}{dx}$

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=1 \\ \Rightarrow \frac{dy}{dx}=\frac{1}{2x^{\frac{1}{2}}}\mathbf{}\mathbf{\left[}\frac{\mathbf{d}}{\mathbf{d}\mathbf{x}}\mathbf{\left(}{\mathbf{x}}^{\mathbf{n}}\mathbf{\right)}\mathbf{=}\mathbf{n}{\mathbf{x}}^{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{\right]} \\ \Rightarrow 1=\frac{1}{2y}\mathbf{[}\sqrt{\mathbf{x}}\mathbf{=}\mathbf{y}\mathbf{]} \\ \Rightarrow y=\frac{1}{2} \end{gathered}$$

Hence, option B is correct.