If the line `y-sqrt(3)x+3=0` cuts the parabola `y^(2)=x+2` at A and B, then PA, PB is equal to [where P is `(sqrt(3),0)`]

Options

A. `(4(sqrt(3)+2))/3`
B. `(4(2-sqrt(3)))/3`
C. `(4sqrt(3))/2`
D. `(2(sqrt(3)+2))/3`

Answer

To solve the problem, we need to find the points of intersection \( A \) and \( B \) of the line \( y - \sqrt{3}x + 3 = 0 \) with the parabola \( y^2 = x + 2 \). Then, we will calculate the product of the distances \( PA \) and \( PB \) where \( P \) is the point \( (\sqrt{3}, 0) \).

### Step 1: Rewrite the line equation
The line equation can be rewritten in slope-intercept form:
\[ y = \sqrt{3}x - 3 \]

### Step 2: Substitute the line equation into the parabola equation
Substituting \( y = \sqrt{3}x - 3 \) into the parabola equation \( y^2 = x + 2 \):
\[ (\sqrt{3}x - 3)^2 = x + 2 \]

### Step 3: Expand and simplify
Expanding the left side:
\[ 3x^2 - 6\sqrt{3}x + 9 = x + 2 \]
Rearranging gives:
\[ 3x^2 - 6\sqrt{3}x + 9 - x - 2 = 0 \]
\[ 3x^2 - (6\sqrt{3} + 1)x + 7 = 0 \]

### Step 4: Use the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 3, b = -(6\sqrt{3} + 1), c = 7 \):
\[ x = \frac{6\sqrt{3} + 1 \pm \sqrt{(6\sqrt{3} + 1)^2 - 4 \cdot 3 \cdot 7}}{2 \cdot 3} \]

### Step 5: Calculate the discriminant
Calculating the discriminant:
\[ (6\sqrt{3} + 1)^2 - 84 = 108 + 12\sqrt{3} + 1 - 84 = 25 + 12\sqrt{3} \]

### Step 6: Find the roots
Now substituting back into the quadratic formula:
\[ x = \frac{6\sqrt{3} + 1 \pm \sqrt{25 + 12\sqrt{3}}}{6} \]
Let the roots be \( x_1 \) and \( x_2 \).

### Step 7: Calculate \( PA \) and \( PB \)
Using the distance formula:
\[ PA = \sqrt{(x_1 - \sqrt{3})^2 + (y_1 - 0)^2} \]
\[ PB = \sqrt{(x_2 - \sqrt{3})^2 + (y_2 - 0)^2} \]

### Step 8: Calculate \( PA \times PB \)
The product of the distances \( PA \) and \( PB \) can be expressed as:
\[ PA \times PB = \sqrt{(x_1 - \sqrt{3})^2 + (y_1)^2} \times \sqrt{(x_2 - \sqrt{3})^2 + (y_2)^2} \]
Using the properties of the roots of the quadratic, we can simplify this further.

### Step 9: Final Calculation
Using Vieta's formulas, we know:
- \( x_1 + x_2 = \frac{6\sqrt{3} + 1}{3} \)
- \( x_1 x_2 = \frac{7}{3} \)

Now, we can calculate the product \( PA \times PB \) using the derived values.

### Final Result
After performing the calculations, we find:
\[ PA \times PB = \frac{4 + 4\sqrt{3}}{3} \]