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

Options

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

Answer

To find the ordinate 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 need to minimize the distance between a point on the curve \( (x, \sqrt{x}) \) and the point \( (2, 1) \). The distance \( D \) can be expressed as:
\[ D = \sqrt{(x - 2)^2 + (\sqrt{x} - 1)^2} \]
To simplify the calculations, we can minimize the square of the distance \( D^2 \):
\[ D^2 = (x - 2)^2 + (\sqrt{x} - 1)^2 \]

### Step 2: Expand the distance squared
Expanding \( D^2 \):
\[ D^2 = (x - 2)^2 + (\sqrt{x} - 1)^2 = (x - 2)^2 + (x - 2\sqrt{x} + 1) \]
\[ = (x - 2)^2 + (x - 2\sqrt{x} + 1) = x^2 - 4x + 4 + x - 2\sqrt{x} + 1 \]
\[ = x^2 - 3x + 5 - 2\sqrt{x} \]

### Step 3: Differentiate the distance squared
Now we differentiate \( D^2 \) with respect to \( x \):
\[ \frac{d(D^2)}{dx} = 2x - 3 - \frac{1}{\sqrt{x}} \]
Setting the derivative equal to zero to find critical points:
\[ 2x - 3 - \frac{1}{\sqrt{x}} = 0 \]
\[ 2x - 3 = \frac{1}{\sqrt{x}} \]
Multiplying through by \( \sqrt{x} \):
\[ 2x\sqrt{x} - 3\sqrt{x} = 1 \]
Rearranging gives:
\[ 2x\sqrt{x} - 3\sqrt{x} - 1 = 0 \]

### Step 4: Substitute \( \sqrt{x} = t \)
Let \( t = \sqrt{x} \), then \( x = t^2 \):
\[ 2t^3 - 3t - 1 = 0 \]

### Step 5: Solve the cubic equation
To find the roots of the cubic equation \( 2t^3 - 3t - 1 = 0 \), we can use the Rational Root Theorem or synthetic division. 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 out \( t + 1 \) from \( 2t^3 - 3t - 1 \).

### Step 6: Perform synthetic division
Dividing \( 2t^3 - 3t - 1 \) by \( t + 1 \):
\[ 2t^3 - 3t - 1 = (t + 1)(2t^2 - 2t - 1) \]
Now we can solve \( 2t^2 - 2t - 1 = 0 \) using the quadratic formula:
\[ t = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} = \frac{2 \pm \sqrt{4 + 8}}{4} = \frac{2 \pm \sqrt{12}}{4} = \frac{2 \pm 2\sqrt{3}}{4} = \frac{1 \pm \sqrt{3}}{2} \]

### Step 7: Find the corresponding \( y \) values
Now, substituting back to find \( y \):
1. For \( t = \frac{1 + \sqrt{3}}{2} \):
\[ y = t = \frac{1 + \sqrt{3}}{2} \]
2. For \( t = \frac{1 - \sqrt{3}}{2} \):
\[ y = t = \frac{1 - \sqrt{3}}{2} \quad \text{(not valid since it will be negative)} \]

### Conclusion
Thus, the ordinate of the point on the curve that is closest to the point \( (2, 1) \) is:
\[ \frac{1 + \sqrt{3}}{2} \]