The line, which is parallel to `X`-axis and crosses the curve `y = sqrtx` at an angle `45^@`, is

Options

A. `y=(1)/(4)`
B. `y=(1)/(2)`
C. `y=1`
D. `y=4`

Answer

To solve the problem, we need to find the equation of a line that is parallel to the x-axis and crosses the curve \( y = \sqrt{x} \) at an angle of \( 45^\circ \).

### Step-by-Step Solution:

1. **Understand the curve**: The given curve is \( y = \sqrt{x} \). This can be rewritten as \( y = x^{1/2} \).

2. **Differentiate the curve**: To find the slope of the tangent to the curve at any point, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}} \]

3. **Set the slope equal to the tangent of the angle**: We know that the line crosses the curve at an angle of \( 45^\circ \). The tangent of \( 45^\circ \) is \( 1 \). Therefore, we set the derivative equal to \( 1 \):
\[ \frac{1}{2\sqrt{x}} = 1 \]

4. **Solve for \( x \)**: Rearranging the equation gives:
\[ 2\sqrt{x} = 1 \implies \sqrt{x} = \frac{1}{2} \implies x = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

5. **Find the corresponding \( y \) value**: Now, we substitute \( x = \frac{1}{4} \) back into the original curve equation to find \( y \):
\[ y = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

6. **Write the equation of the line**: A line that is parallel to the x-axis has the form \( y = k \). Since we found \( y = \frac{1}{2} \), the equation of the line is:
\[ y = \frac{1}{2} \]

### Final Answer:
The line that is parallel to the x-axis and crosses the curve \( y = \sqrt{x} \) at an angle of \( 45^\circ \) is:
\[ y = \frac{1}{2} \]