Find the equation of the hyperbola whose

vertices are (-8, -1) and (16, -1) and focus is (17, -1)

Given: Vertices are (-8, -1) and (16, -1) and focus is (17, -1)

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

Center of hyperbola having vertices (-8, -1) and (16, -1) is given by

= (4, -1)

The distance between two vertices is 2a and vertices are (-8, -1) and (16, -1)

The distance between the foci and vertex is ae – a, Foci is (17, -1) and the vertex is (16, -1)

b² = a²(e² – 1)

The equation of hyperbola:

⇒ 25(x² + 16 – 8x) – 144(y² + 1 + 2y) = 3600

⇒ 25x² + 400 – 200x – 144y² – 144 – 288y – 3600 = 0

⇒ 25x² – 144y² – 200x – 288y – 3344 = 0

Hence, required equation of hyperbola is 25x² – 144y² – 200x – 288y – 3344 = 0