The shortest distance between the point `(0, 1/2)` and the curve ` x = sqrt(y) , (y gt 0 ) ` , is:

Options

A. `3/2`
B. `(sqrt5)/(2)`
C. `(sqrt3)/(2)`
D. `1/2`

Explanation

To find the shortest distance between the point \( (0, \frac{1}{2}) \) and the curve \( x = \sqrt{y} \) (where \( y > 0 \)), we can follow these steps:

### Step 1: Parameterize the curve
The curve is given by \( x = \sqrt{y} \). We can express \( y \) in terms of \( x \):
\[ y = x^2 \]
This means any point on the curve can be represented as \( (x, x^2) \).

### Step 2: Define the distance function
We need to find the distance \( d \) between the point \( (0, \frac{1}{2}) \) and a point \( (x, x^2) \) on the curve. The distance \( d \) is given by:
\[ d = \sqrt{(x - 0)^2 + \left(x^2 - \frac{1}{2}\right)^2} \]
Simplifying this, we have:
\[ d = \sqrt{x^2 + \left(x^2 - \frac{1}{2}\right)^2} \]

### Step 3: Simplify the distance function
To minimize the distance, we can minimize \( d^2 \) instead:
\[ d^2 = x^2 + \left(x^2 - \frac{1}{2}\right)^2 \]
Expanding the second term:
\[ d^2 = x^2 + (x^4 - x^2 + \frac{1}{4}) = x^4 + \frac{1}{4} \]
Thus, we have:
\[ d^2 = x^4 + \frac{1}{4} \]

### Step 4: Find the minimum distance
To find the minimum distance, we need to find the critical points of \( d^2 \). Since \( d^2 \) is a function of \( x \), we can take the derivative and set it to zero:
\[ \frac{d(d^2)}{dx} = 4x^3 \]
Setting the derivative equal to zero gives:
\[ 4x^3 = 0 \implies x = 0 \]

### Step 5: Calculate the minimum distance
Substituting \( x = 0 \) back into the distance function:
\[ d^2 = (0)^4 + \frac{1}{4} = \frac{1}{4} \]
Thus, the minimum distance \( d \) is:
\[ d = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

### Conclusion
The shortest distance between the point \( (0, \frac{1}{2}) \) and the curve \( x = \sqrt{y} \) is:
\[ \boxed{\frac{1}{2}} \]