Question

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

Answer 1

Given, x² - 3y² - 4x - 6y - 11 = 0
⇒ x² - 4x + 4 - 3y² - 6y - 3 - 12 = 0
⇒ (x - 2)² -3(y² + 2y + 1) = 12
⇒ (x - 2)² - 3(y + 1 )² = 12
⇒ (x - 2)²/12 - (y + 1)²/4 = 1
⇒(x - 2)²/(2√3)² - (y + 1)²/2² = 1
Compare this with standard form e.g., x²/a² - y²/b² = 1
well, (2, -1) is the centre of hyperbola .
a² = 12 and b² = 4

We know, c² = a² + b² , here c is the distance of foci and centre of hyperbola.
c² = 12 + 4 = 16
c = ±4
Hence, foci (2 ± 4, -1)
Distance between foci = |2c| = 2×4 = 8 unit

Hence answer is 8 unit