Answer to Question #104116 in Analytic Geometry for Deepak Rana

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

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

the equation of the hyperbola will be

$(y-k)^2/a^2 - (x-h)/b^2=1 \newline (y-0)^2/16 -(x-1)/20 =1$