SOLUTION: Write an equation for the hyperbola with center at ​(4​,-6​), focus at ​(7​,-6​), and vertex at (6,-6) An equation for the hyperbola is_____. (Simplify your answer. T

Algebra ->  Equations -> SOLUTION: Write an equation for the hyperbola with center at ​(4​,-6​), focus at ​(7​,-6​), and vertex at (6,-6) An equation for the hyperbola is_____. (Simplify your answer. T      Log On


   

Question 1194464: Write an equation for the hyperbola with center at ​(4​,-6​), focus at ​(7​,-6​), and vertex at (6,-6)
An equation for the hyperbola is_____.
(Simplify your answer. Type your answer in standard form. Use integers or fractions for any numbers in the​ equation.)


Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
The equation of a hyperbola is
%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=1, where (h,k) is the center, a and b are the lengths of the semi-major and the semi-minor axes.
given:
center at ​(4​,-6​)=>h=4, k+=+-6
focus at ​(7​,-6%E2%80%8B) =(h%2Bc,k)=>h=4, 4%2Bc=7=>c=7-4=3
vertex at (6,-6)=(h%2Ba,k)=>h=4,4%2Ba=6=>a=2
b%5E2=c%5E2-a%5E2
b%5E2=3%5E2-2%5E2
b%5E2=9-4
b%5E2=5
b=sqrt%285%29
The equation of a hyperbola is
%28x-4%29%5E2%2F2%5E2-%28y-%28-6%29%29%5E2%2F%28sqrt%285%29%29%5E2=1
%28x-4%29%5E2%2F4-%28y%2B6%29%5E2%2F5=1