What is the standard form equation of the hyperbola that has

What is the standard form equation of the hyperbola that has vertices(1,2)(1,2)and(1,8)(1,8)and foci(1,10)(1,10)and(1,16)?

Solution

vertices (1,-2) ,(1,8)

focii(1,-10), (1,16)

This hyperbola is of the form

(y-k)²/a² - (x-h)²/b²=1

Centre is (h,k)

Vertices (h,k+-a)

h=1,       k+a= 8   and k-a=-2

k=3          a= 5

focii (h,k+-c)

k+c=16   and k-c=-10

c=13

c^2=a^2 + b^2

169-25=b^2

b=12

The required equation is

(y-3)²/25 - (x-1)²/144=1