Find the equation of the hyperbola whose

the focus is at (5, 2), vertices at (4, 2) and (2, 2) and centre at (3, 2)

Given: Vertices are (4, 2) and (2, 2), the focus is (5, 2) and centre (3, 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

The distance between two vertices is 2a and vertices are (4, 2) and (2, 2)

The distance between the foci and vertex is ae – a, Foci is (5, 2) and the vertex is (4, 2)

⇒ 1 = e – 1

⇒ e = 1 + 1

⇒ e = 2

b² = a²(e² – 1)

The equation of hyperbola:

{∵ Centre (3, 2)}

⇒ 3(x² + 9 – 6x) – (y² + 4 – 4y) = 3

⇒ 3x² + 27 – 18x – y² – 4 + 4y – 3 = 0

⇒ 3x² – y² – 18x + 4y + 20 = 0

Hence, required equation of hyperbola is 3x² – y² – 18x + 4y + 20 = 0