Question

Find the equation of the hyperbola with vertices (1,−4) and (1,4), and foci at(1,−6) and (1,6).

Answer 1

One thing to notice in all the coordinates is that their x coordinate are same i.e. all the points lie on a line x=1

hence the center of the hyperbola = (1,(-4+4)/2) = (1,0)

distance between center and vertex (a) = 4

the foci are at distance 6 from the center , hence c=6

$a^2+b^2=c^2 \\b^2=20$

$$\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$$

$$\frac{(y-0)^2}{16}-\frac{(x-1)^2}{20}=1$$