Given: Foci are (4, 2) and (8, 2) and eccentricity is 2

To find: equation of the hyperbola

Formula used:

The standard form of the equation of the hyperbola is,

Center is the mid-point of two foci.

Distance between the foci is 2ae and b² = a²(e² – 1)

The distance between two points (m, n) and (a, b) is given by

Mid-point theorem:

Mid-point of two points (m, n) and (a, b) is given by

Center of hyperbola having foci (4, 2) and (8, 2) is given by

= (6, 2)

The distance between the foci is 2ae and Foci are (4, 2) and (8, 2)

{∵ e = 2}

b² = a²(e² – 1)

The equation of hyperbola:

⇒ 3(x² + 36 – 12x) – (y² + 4 – 4y) = 3

⇒ 3x² + 108 – 36x – y² – 4 + 4y – 3 = 0

⇒ 3x² – y² – 36x + 4y + 101 = 0

Hence, required equation of hyperbola is 3x² – y² – 36x + 4y + 101 = 0