Statement-1: The point `((1)/(4), (1)/(2))` on the parabola `y^(2)=x` is closest to the line `y=x+1`
Statement-2: The tangent at `((1)/(4), (1)/(2))` to the parabola `y^(2)=x` is parallel to the line `y=x+1`

Options

A. Statement-1 is true, statement-2 is true, statement-2 is not a correct explanation for statement-1
B. Statement-1 is true, statement-2 is false. S
C. Statement-1 is false, statement-2 is true.
D. Statement-1 is true, statement-2 is true.
statement-2 is a correct explanation for statement-1.

Answer

To solve the problem, we need to analyze the statements given regarding the point on the parabola and the line.

### Step 1: Identify the given elements
We have a parabola given by the equation:
\[ y^2 = x \]
And a line given by the equation:
\[ y = x + 1 \]
We need to check the validity of the two statements regarding the point \(\left(\frac{1}{4}, \frac{1}{2}\right)\).

### Step 2: Find the slope of the line
The slope of the line \(y = x + 1\) is \(1\).

### Step 3: Find the equation of the tangent to the parabola at the given point
For the parabola \(y^2 = x\), the slope of the tangent at any point \((x_0, y_0)\) can be derived from the implicit differentiation:
1. Differentiate \(y^2 = x\) to find \(\frac{dy}{dx}\):
\[ 2y \frac{dy}{dx} = 1 \implies \frac{dy}{dx} = \frac{1}{2y} \]
2. At the point \(\left(\frac{1}{4}, \frac{1}{2}\right)\), substitute \(y = \frac{1}{2}\):
\[ \frac{dy}{dx} = \frac{1}{2 \cdot \frac{1}{2}} = 1 \]

### Step 4: Write the equation of the tangent line
Using the point-slope form of the line equation:
\[ y - y_0 = m(x - x_0) \]
Substituting \(m = 1\), \(x_0 = \frac{1}{4}\), and \(y_0 = \frac{1}{2}\):
\[ y - \frac{1}{2} = 1\left(x - \frac{1}{4}\right) \]
This simplifies to:
\[ y = x + \frac{1}{4} \]

### Step 5: Compare the slopes of the tangent and the line
The slope of the tangent line is \(1\), which is the same as the slope of the line \(y = x + 1\). Thus, the tangent at the point \(\left(\frac{1}{4}, \frac{1}{2}\right)\) is parallel to the line \(y = x + 1\).

### Step 6: Determine the closest point on the parabola to the line
Since the tangent at \(\left(\frac{1}{4}, \frac{1}{2}\right)\) is parallel to the line \(y = x + 1\), this point is indeed the closest point on the parabola to the line.

### Conclusion
Both statements are true:
- Statement 1 is true: The point \(\left(\frac{1}{4}, \frac{1}{2}\right)\) on the parabola is the closest to the line \(y = x + 1\).
- Statement 2 is true: The tangent at \(\left(\frac{1}{4}, \frac{1}{2}\right)\) to the parabola is parallel to the line \(y = x + 1\).

### Final Answer
Both statements are true.

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