Write in standard form of the hyperbola with the veritces (9,1) and (11,1) and foci at (8,1) and (12,1)

Question

Write in standard form

Answer 1

Nick, since the y-coordinates of the vertices and foci are the same indicates this is a horizontal hyperbola. The equation of the hyperbola is in the form:

 

(x-h)^2/a^2-(y-k)^2/b^2=1

 

The distance between the vertices is 2a=11-9=2

 

2a=2  Divide both sides of the equation by 2. This implies a=1

 

The center of the hyperbola is ((9+11)/2,(1+1)/2)=(20/2,2/2)=(10,1)  This implies h=10 and k=1.

 

Find the distance from the center to the foci in which is c=12-10=2

 

Determine b^2 from the equation c^2=a^2+b^2

 

b^2=c^2-a^2=2^2-1^2=4-1=3

 

Substitute these into the hyperbola equation in which gives its equation based upon the given conditions stated in the problem.

 

(x-10)^2/1-(y-1)^2/3=1