The shortest distance between the point `((3)/(2),0)` and the curve `y=sqrt(x),(x gt 0)`, is

Options

A. `sqrt5/2`
B. `5/4`
C. `3/2`
D. `sqrt3/2`

Answer

To find the shortest distance between the point \((\frac{3}{2}, 0)\) and the curve \(y = \sqrt{x}\) (where \(x > 0\)), we can follow these steps:

### Step 1: Define the points
Let the point on the curve be \((h, \sqrt{h})\), where \(h > 0\). The point we are considering is \((\frac{3}{2}, 0)\).

### Step 2: Write the distance formula
The distance \(d\) between the point \((\frac{3}{2}, 0)\) and the point \((h, \sqrt{h})\) is given by the formula:
\[ d = \sqrt{(h - \frac{3}{2})^2 + (\sqrt{h} - 0)^2} \]

### Step 3: Simplify the distance formula
Squaring the distance (to simplify calculations) gives:
\[ d^2 = (h - \frac{3}{2})^2 + (\sqrt{h})^2 \]
\[ = (h - \frac{3}{2})^2 + h \]

### Step 4: Expand the expression
Now, we expand the squared term:
\[ d^2 = (h^2 - 3h + \frac{9}{4}) + h \]
\[ = h^2 - 2h + \frac{9}{4} \]

### Step 5: Rewrite the expression
We can rewrite this expression:
\[ d^2 = h^2 - 2h + 1 + \frac{5}{4} \]
\[ = (h - 1)^2 + \frac{5}{4} \]

### Step 6: Find the minimum distance
The term \((h - 1)^2\) is always non-negative and reaches its minimum value of \(0\) when \(h = 1\). Therefore, the minimum value of \(d^2\) occurs at:
\[ d^2 = 0 + \frac{5}{4} = \frac{5}{4} \]

### Step 7: Calculate the shortest distance
To find the shortest distance \(d\), we take the square root:
\[ d = \sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2} \]

### Final Answer
Thus, the shortest distance between the point \((\frac{3}{2}, 0)\) and the curve \(y = \sqrt{x}\) is:
\[ \frac{\sqrt{5}}{2} \]

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