Distance of point P on the curve `y=x^(3//2)` which is nearest to the point M (4, 0) from origin is

Options

A. `sqrt((112)/(27))`
B. `sqrt((100)/(27))`
C. `sqrt((101)/(9))`
D. `sqrt((112)/(9))`

Explanation

To solve the problem of finding the distance of the point P on the curve \( y = x^{3/2} \) that is nearest to the point M(4, 0) from the origin, we can follow these steps:

### Step 1: Define the distance function
We need to find the distance from point P on the curve to point M. The coordinates of point P can be expressed as \( P(x, y) = (x, x^{3/2}) \). The distance \( D \) from point P to point M(4, 0) is given by the distance formula:
\[ D = \sqrt{(x - 4)^2 + (x^{3/2} - 0)^2} \]
This simplifies to:
\[ D = \sqrt{(x - 4)^2 + (x^{3/2})^2} = \sqrt{(x - 4)^2 + x^3} \]

### Step 2: Minimize the distance
To minimize the distance \( D \), we can minimize \( D^2 \) instead (to avoid dealing with the square root):
\[ D^2 = (x - 4)^2 + x^3 \]

### Step 3: Differentiate \( D^2 \)
Now, we differentiate \( D^2 \) with respect to \( x \):
\[ \frac{d(D^2)}{dx} = 2(x - 4) + 3x^2 \]
Setting the derivative equal to zero to find critical points:
\[ 2(x - 4) + 3x^2 = 0 \]
This simplifies to:
\[ 3x^2 + 2x - 8 = 0 \]

### Step 4: Solve the quadratic equation
We can solve the quadratic equation \( 3x^2 + 2x - 8 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 3, b = 2, c = -8 \):
\[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 3 \cdot (-8)}}{2 \cdot 3} = \frac{-2 \pm \sqrt{4 + 96}}{6} = \frac{-2 \pm \sqrt{100}}{6} = \frac{-2 \pm 10}{6} \]
This gives us two solutions:
\[ x = \frac{8}{6} = \frac{4}{3} \quad \text{and} \quad x = \frac{-12}{6} = -2 \]
Since \( x = -2 \) is not valid for our curve (as \( y = x^{3/2} \) is not defined for negative \( x \)), we take \( x = \frac{4}{3} \).

### Step 5: Find the corresponding y-coordinate
Now we find the corresponding y-coordinate:
\[ y = \left(\frac{4}{3}\right)^{3/2} = \frac{8}{3\sqrt{3}} = \frac{8\sqrt{3}}{9} \]
So the point P is \( P\left(\frac{4}{3}, \frac{8\sqrt{3}}{9}\right) \).

### Step 6: Calculate the distance from the origin
Now we calculate the distance from the origin (0, 0) to point P:
\[ OP = \sqrt{\left(\frac{4}{3}\right)^2 + \left(\frac{8\sqrt{3}}{9}\right)^2} \]
Calculating each term:
\[ \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]
\[ \left(\frac{8\sqrt{3}}{9}\right)^2 = \frac{64 \cdot 3}{81} = \frac{192}{81} = \frac{64}{27} \]
Now, adding these:
\[ OP^2 = \frac{16}{9} + \frac{64}{27} \]
To add these fractions, we need a common denominator:
\[ \frac{16}{9} = \frac{48}{27} \]
Thus,
\[ OP^2 = \frac{48}{27} + \frac{64}{27} = \frac{112}{27} \]
Finally, the distance \( OP \) is:
\[ OP = \sqrt{\frac{112}{27}} = \frac{\sqrt{112}}{\sqrt{27}} = \frac{4\sqrt{7}}{3\sqrt{3}} = \frac{4\sqrt{21}}{9} \]

### Final Answer
The distance of point P from the origin is:
\[ \frac{4\sqrt{21}}{9} \]