If the lie `y=xsqrt(3)` cuts the curve `x^(3)+y^(3)+3xy+5x^(2)+3y^(2)+4x+5y+1=0` at the points A,B,C then OA.OB.OC is equal to

Options

A. `4/13(3sqrt(3)-1)`
B. `3sqrt(3)+1`
C. `1/(sqrt(3)(2+7sqrt(3))`
D. None of these

Answer

To solve the problem, we need to find the product of the distances from the origin \( O \) to the points \( A, B, C \) where the line \( y = x\sqrt{3} \) intersects the curve given by the equation:

\[ x^3 + y^3 + 3xy + 5x^2 + 3y^2 + 4x + 5y + 1 = 0 \]

### Step 1: Substitute the line equation into the curve equation

We start by substituting \( y = x\sqrt{3} \) into the curve equation.

\[ x^3 + (x\sqrt{3})^3 + 3x(x\sqrt{3}) + 5x^2 + 3(x\sqrt{3})^2 + 4x + 5(x\sqrt{3}) + 1 = 0 \]

This simplifies to:

\[ x^3 + 3\sqrt{3}x^3 + 3\sqrt{3}x^2 + 5x^2 + 9x^2 + 4x + 5\sqrt{3}x + 1 = 0 \]

Combining like terms, we have:

\[ (1 + 3\sqrt{3})x^3 + (3\sqrt{3} + 5 + 9)x^2 + (4 + 5\sqrt{3})x + 1 = 0 \]

### Step 2: Identify the coefficients

Let’s denote the coefficients:

- Coefficient of \( x^3 \): \( a = 1 + 3\sqrt{3} \)
- Coefficient of \( x^2 \): \( b = 3\sqrt{3} + 14 \)
- Coefficient of \( x \): \( c = 4 + 5\sqrt{3} \)
- Constant term: \( d = 1 \)

### Step 3: Use Vieta's formulas

According to Vieta's formulas, the product of the roots \( x_1, x_2, x_3 \) of the cubic equation \( ax^3 + bx^2 + cx + d = 0 \) is given by:

\[ x_1 x_2 x_3 = -\frac{d}{a} \]

Substituting the values we found:

\[ x_1 x_2 x_3 = -\frac{1}{1 + 3\sqrt{3}} \]

### Step 4: Calculate \( OA \cdot OB \cdot OC \)

The distances from the origin to the points \( A, B, C \) can be expressed as:

\[ OA = \sqrt{x_1^2 + (x_1\sqrt{3})^2} = \sqrt{x_1^2 + 3x_1^2} = 2|x_1| \]

Similarly,

\[ OB = 2|x_2| \quad \text{and} \quad OC = 2|x_3| \]

Thus, the product \( OA \cdot OB \cdot OC \) is:

\[ OA \cdot OB \cdot OC = (2|x_1|)(2|x_2|)(2|x_3|) = 8 |x_1 x_2 x_3| \]

Substituting the value from Vieta's:

\[ OA \cdot OB \cdot OC = 8 \left| -\frac{1}{1 + 3\sqrt{3}} \right| = \frac{8}{1 + 3\sqrt{3}} \]

### Final Answer

Thus, the value of \( OA \cdot OB \cdot OC \) is:

\[ \frac{8}{1 + 3\sqrt{3}} \]