The x co-ordinate of the point on the curve `y = sqrtx` which is closest to the point `(2,1)` is :

Options

A. `(2+ sqrt3)/(2)`
B. `(1+sqrt2)/(2)`
C. `(-1+ sqrt3)/(2)`
D. `1`

Answer

To find the x-coordinate of the point on the curve \( y = \sqrt{x} \) that is closest to the point \( (2, 1) \), we can follow these steps:

### Step 1: Define the distance function
We want to minimize the distance \( D \) between a point \( P(x, y) \) on the curve and the point \( Q(2, 1) \). The point on the curve can be represented as \( P(t^2, t) \), where \( y = \sqrt{x} \) implies \( x = t^2 \) (with \( t \geq 0 \)). The distance \( D \) is given by:
\[ D = \sqrt{(t^2 - 2)^2 + (t - 1)^2} \]

### Step 2: Simplify the distance function
To simplify the calculations, we minimize the square of the distance \( D^2 \):
\[ D^2 = (t^2 - 2)^2 + (t - 1)^2 \]
Expanding this:
\[ D^2 = (t^4 - 4t^2 + 4) + (t^2 - 2t + 1) = t^4 - 3t^2 - 2t + 5 \]

### Step 3: Find the derivative
To find the minimum distance, we take the derivative of \( D^2 \) with respect to \( t \) and set it to zero:
\[ \frac{d(D^2)}{dt} = 4t^3 - 6t - 2 \]
Setting the derivative equal to zero:
\[ 4t^3 - 6t - 2 = 0 \]

### Step 4: Solve the cubic equation
Dividing the entire equation by 2:
\[ 2t^3 - 3t - 1 = 0 \]
We can use the Rational Root Theorem or synthetic division to find the roots. Testing \( t = 1 \):
\[ 2(1)^3 - 3(1) - 1 = 2 - 3 - 1 = -2 \quad \text{(not a root)} \]
Testing \( t = -1 \):
\[ 2(-1)^3 - 3(-1) - 1 = -2 + 3 - 1 = 0 \quad \text{(is a root)} \]
Now we can factor \( (t + 1) \) out of \( 2t^3 - 3t - 1 \).

### Step 5: Factor the cubic polynomial
Using synthetic division, we find:
\[ 2t^3 - 3t - 1 = (t + 1)(2t^2 - 2t - 1) \]
Now we solve \( 2t^2 - 2t - 1 = 0 \) using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} = \frac{2 \pm \sqrt{4 + 8}}{4} = \frac{2 \pm \sqrt{12}}{4} = \frac{2 \pm 2\sqrt{3}}{4} = \frac{1 \pm \sqrt{3}}{2} \]
Since \( t \) must be positive, we take:
\[ t = \frac{1 + \sqrt{3}}{2} \]

### Step 6: Find the x-coordinate
Now, we find the x-coordinate:
\[ x = t^2 = \left(\frac{1 + \sqrt{3}}{2}\right)^2 = \frac{(1 + \sqrt{3})^2}{4} = \frac{1 + 2\sqrt{3} + 3}{4} = \frac{4 + 2\sqrt{3}}{4} = 1 + \frac{\sqrt{3}}{2} \]

### Final Answer
Thus, the x-coordinate of the point on the curve \( y = \sqrt{x} \) which is closest to the point \( (2, 1) \) is:
\[ \boxed{1 + \frac{\sqrt{3}}{2}} \]