Given: Foci is (4, 2), e = 2 and center at (6, 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 vertices

The distance between two vertices is 2a

The distance between the foci and vertex is ae – a 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

Therefore

Let one of the two foci is (m, n) and the other one is (4, 2)

Since, Centre(6, 2)

Foci are (4, 2) and (8, 2)

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

⇒ a² = 1

b² = a²(e² – 1)

⇒ b² = 4 – 1

⇒ b² = 3

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