Answer :

**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 b^{2} = a^{2}(e^{2} – 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)

b^{2} = a^{2}(e^{2} – 1)

The equation of hyperbola:

⇒ 25(x^{2} + 16 – 8x) – 144(y^{2} + 1 + 2y) = 3600

⇒ 25x^{2} + 400 – 200x – 144y^{2} – 144 – 288y – 3600 = 0

⇒ 25x^{2} – 144y^{2} – 200x – 288y – 3344 = 0

Hence, required equation of hyperbola is **25x ^{2} – 144y^{2} – 200x – 288y – 3344 = 0**

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