How do you find the standard form of the equation of the hyperbola given the properties vertices (3,2), (13,2); endpoints of the conjugate axis (8,4), (8,0)?

1 Answer
May 17, 2017

Answer:

Equation of hyperbola is #(x-8)^2/25-(y-2)^2/4=1#

Explanation:

As the ordinates of vertices are equal, the equation of hyperbola is of the form #(x-h)^2/a^2-(y-k)^2/b^2=1#, where #(h,k)# is the centre of hyperbola. As vertices are #(3,2)# and #(13,2)#, centre is #((3+13)/2,(2+2)/2)# i.e. #(8,2)# and hence equation of hyperbola is

#(x-8)^2/a^2-(y-2)^2/b^2=1#

To find #a# and #b#, we will work them out using their relation with vertices and endpoints of conjugate axis. Now length of conjugate axis is #2b# and as the distance between endpoints is #4#, we have #2b=4# i.e. #b=2#.

We have the distance between vertices as #10# and as this is #2a#, we have #a=5# and hence equation of hyperbola is

#(x-8)^2/25-(y-2)^2/4=1#

graph{((x-8)^2/25-(y-2)^2/4-1)((x-13)^2+(y-2)^2-0.03)((x-3)^2+(y-2)^2-0.03)=0 [-12, 28, -7.92, 12.08]}