If `(alpha , beta)` is the point on `y^2=6x` , that is closest to `(3,3/2)` then find the value of `2(alpha+beta)`

Answer

To find the point \((\alpha, \beta)\) on the curve \(y^2 = 6x\) that is closest to the point \((3, \frac{3}{2})\), we can follow these steps:

### Step 1: Express the relationship between \(\alpha\) and \(\beta\)
Since the point \((\alpha, \beta)\) lies on the curve \(y^2 = 6x\), we can express this relationship as:
\[ \beta^2 = 6\alpha \]

### Step 2: Write the distance squared between the points
The distance \(d\) between the points \((\alpha, \beta)\) and \((3, \frac{3}{2})\) can be expressed using the distance formula. To simplify our calculations, we will work with the square of the distance \(d^2\):
\[ d^2 = (\alpha - 3)^2 + \left(\beta - \frac{3}{2}\right)^2 \]

### Step 3: Expand the distance squared
Expanding the expression for \(d^2\):
\[ d^2 = (\alpha - 3)^2 + \left(\beta - \frac{3}{2}\right)^2 = (\alpha^2 - 6\alpha + 9) + \left(\beta^2 - 3\beta + \frac{9}{4}\right) \]
Substituting \(\beta^2 = 6\alpha\) into the equation:
\[ d^2 = \alpha^2 - 6\alpha + 9 + (6\alpha - 3\beta + \frac{9}{4}) \]
This simplifies to:
\[ d^2 = \alpha^2 - 3\beta + 9 + \frac{9}{4} \]

### Step 4: Substitute \(\beta\) in terms of \(\alpha\)
From \(\beta^2 = 6\alpha\), we can express \(\beta\) as:
\[ \beta = \sqrt{6\alpha} \]
Now substituting this into \(d^2\):
\[ d^2 = \alpha^2 - 3\sqrt{6\alpha} + \frac{45}{4} \]

### Step 5: Differentiate to find the minimum distance
To find the minimum distance, we differentiate \(d^2\) with respect to \(\alpha\) and set the derivative to zero:
\[ \frac{d(d^2)}{d\alpha} = 2\alpha - \frac{3 \cdot 6}{2\sqrt{6\alpha}} = 0 \]
This simplifies to:
\[ 2\alpha - \frac{9}{\sqrt{6\alpha}} = 0 \]

### Step 6: Solve for \(\alpha\)
Rearranging gives:
\[ 2\alpha\sqrt{6\alpha} = 9 \]
Squaring both sides:
\[ 4\alpha^2 \cdot 6\alpha = 81 \implies 24\alpha^3 = 81 \implies \alpha^3 = \frac{81}{24} = \frac{27}{8} \]
Taking the cube root:
\[ \alpha = \left(\frac{27}{8}\right)^{\frac{1}{3}} = \frac{3}{2} \]

### Step 7: Find \(\beta\)
Substituting \(\alpha = \frac{3}{2}\) back to find \(\beta\):
\[ \beta = \sqrt{6 \cdot \frac{3}{2}} = \sqrt{9} = 3 \]

### Step 8: Calculate \(2(\alpha + \beta)\)
Now, we calculate:
\[ 2(\alpha + \beta) = 2\left(\frac{3}{2} + 3\right) = 2\left(\frac{3}{2} + \frac{6}{2}\right) = 2\left(\frac{9}{2}\right) = 9 \]

Thus, the final answer is:
\[ \boxed{9} \]