The distance between the foci of the hyperbola x² - 3y² - 4x - 6y - 11 = 0 is

• A. 4
• B. 6
• C. 8
• D. 10

To find the distance between the foci of the hyperbola given by the equation x² - 3y² - 4x - 6y - 11 = 0, we first need to rewrite it in standard form.

Rearranging the Equation

Start by grouping the x and y terms:

x² - 4x and -3y² - 6y.

Completing the Square

• For the x terms: x² - 4x can be rewritten as (x - 2)² - 4.
• For the y terms: -3(y² + 2y) can be rewritten as -3((y + 1)² - 1) = -3(y + 1)² + 3.

Putting It All Together

Substituting these back into the equation gives:

(x - 2)² - 4 - 3(y + 1)² + 3 - 11 = 0.

This simplifies to:

(x - 2)² - 3(y + 1)² = 12.

Standard Form of the Hyperbola

Dividing everything by 12, we get:

(x - 2)²/12 - (y + 1)²/4 = 1.

Identifying Parameters

From this equation, we can identify:

• a² = 12 (so a = √12 = 2√3)
• b² = 4 (so b = 2)

Calculating the Distance Between Foci

The distance between the foci of a hyperbola is given by the formula 2c, where c = √(a² + b²).

Calculating c:

c = √(12 + 4) = √16 = 4.

Thus, the distance between the foci is:

2c = 2 * 4 = 8.

Final Answer

The distance between the foci of the hyperbola is 8, so the correct choice is C. 8.