If the line `y = x` cuts the curve `x^(3) + 3y^(3) - 30xy + 72x - 55 = 0` in points A,B and C, then the value of`(4sqrt(2))/(55) OA.OB.OC` (where O is the origin), is

Options

A. 55
B. `1/(4sqrt(2))`
C. 2
D. 4

Answer

To solve the problem, we need to find the points of intersection of the line \( y = x \) with the curve given by the equation:

\[ x^3 + 3y^3 - 30xy + 72x - 55 = 0 \]

### Step 1: Substitute \( y = x \) into the curve equation

We substitute \( y = x \) into the curve equation:

\[ x^3 + 3(x^3) - 30x^2 + 72x - 55 = 0 \]

This simplifies to:

\[ 4x^3 - 30x^2 + 72x - 55 = 0 \]

### Step 2: Find the roots of the cubic equation

We need to solve the cubic equation:

\[ 4x^3 - 30x^2 + 72x - 55 = 0 \]

Let the roots of this cubic equation be \( x_1, x_2, x_3 \).

### Step 3: Use Vieta's formulas to find the product of the roots

According to Vieta's formulas, for a cubic equation of the form \( ax^3 + bx^2 + cx + d = 0 \):

- The sum of the roots \( x_1 + x_2 + x_3 = -\frac{b}{a} \)
- The sum of the product of the roots taken two at a time \( x_1x_2 + x_2x_3 + x_3x_1 = \frac{c}{a} \)
- The product of the roots \( x_1x_2x_3 = -\frac{d}{a} \)

For our equation \( 4x^3 - 30x^2 + 72x - 55 = 0 \):

- \( a = 4 \)
- \( b = -30 \)
- \( c = 72 \)
- \( d = -55 \)

Thus, the product of the roots is:

\[ x_1x_2x_3 = -\frac{-55}{4} = \frac{55}{4} \]

### Step 4: Calculate the distances \( OA, OB, OC \)

The distances from the origin \( O(0, 0) \) to the points \( A, B, C \) are given by:

\[ OA = \sqrt{x_1^2 + x_1^2} = x_1\sqrt{2} \]
\[ OB = \sqrt{x_2^2 + x_2^2} = x_2\sqrt{2} \]
\[ OC = \sqrt{x_3^2 + x_3^2} = x_3\sqrt{2} \]

### Step 5: Calculate \( OA \cdot OB \cdot OC \)

Now, we can find:

\[ OA \cdot OB \cdot OC = (x_1\sqrt{2}) \cdot (x_2\sqrt{2}) \cdot (x_3\sqrt{2}) = \sqrt{2}^3 \cdot (x_1 x_2 x_3) = 2\sqrt{2} \cdot \frac{55}{4} \]

### Step 6: Substitute into the final expression

Now we substitute this into the expression \( \frac{4\sqrt{2}}{55} OA \cdot OB \cdot OC \):

\[ \frac{4\sqrt{2}}{55} \cdot 2\sqrt{2} \cdot \frac{55}{4} \]

### Step 7: Simplify the expression

This simplifies to:

\[ \frac{4\sqrt{2} \cdot 2\sqrt{2} \cdot 55}{55 \cdot 4} = \frac{8 \cdot 2}{4} = 4 \]

Thus, the final answer is:

\[ \boxed{4} \]