The point on the curve `y^(2) = x` , where tangent make an angle of `(pi)/(4)` with the x-axis, is

Options

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

Answer

To solve the problem, we need to find the point on the curve \( y^2 = x \) where the tangent makes an angle of \( \frac{\pi}{4} \) with the x-axis.

### Step-by-Step Solution:

1. **Differentiate the curve**:
We start with the equation of the curve:
\[ y^2 = x \]
We differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(x) \]
Using the chain rule, we get:
\[ 2y \frac{dy}{dx} = 1 \]

2. **Solve for \(\frac{dy}{dx}\)**:
Rearranging the equation gives:
\[ \frac{dy}{dx} = \frac{1}{2y} \]

3. **Find the slope of the tangent**:
The slope of the tangent line \( m \) is given by:
\[ m = \frac{dy}{dx} \]
Since the tangent makes an angle of \( \frac{\pi}{4} \) with the x-axis, we have:
\[ m = \tan\left(\frac{\pi}{4}\right) = 1 \]

4. **Set the two expressions for the slope equal**:
Now we set the expressions for the slope equal to each other:
\[ \frac{1}{2y} = 1 \]

5. **Solve for \( y \)**:
Multiplying both sides by \( 2y \) gives:
\[ 1 = 2y \]
Thus, we find:
\[ y = \frac{1}{2} \]

6. **Find the corresponding \( x \)**:
Substitute \( y = \frac{1}{2} \) back into the curve equation to find \( x \):
\[ y^2 = x \implies \left(\frac{1}{2}\right)^2 = x \implies x = \frac{1}{4} \]

7. **Conclusion**:
The point on the curve where the tangent makes an angle of \( \frac{\pi}{4} \) with the x-axis is:
\[ \left(\frac{1}{4}, \frac{1}{2}\right) \]

### Final Answer:
The point is \( \left(\frac{1}{4}, \frac{1}{2}\right) \).