SOLUTION: The asymptotes of a hyperbola are y=2x and y=-2x. If the hyperbola passes through the point (9, 16), find the x-intercepts of the hyperbola.

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: The asymptotes of a hyperbola are y=2x and y=-2x. If the hyperbola passes through the point (9, 16), find the x-intercepts of the hyperbola.      Log On


   

Question 1065505: The asymptotes of a hyperbola are y=2x and y=-2x. If the hyperbola passes through the point (9, 16), find the x-intercepts of the hyperbola.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
A hyperbola that has x-intercepts has an equation of the form
x%5E2%2Fa%5E2-y%2Fb%5E2=1 , and its asymptotes are
y=%28b%2Fa%29%2Ax and y=-%28b%2Fa%29%2Ax ,
so for this problem
b%2Fa=2 <---> b=2a .
That makes the equation
x%5E2%2Fa%5E2-y%5E2%2F4a%5E2=1 so far.
To find the value for a%5E2 ,
we substitute system%28x=9%2Cy=16%29 ,
the coordinates of point (9,16) .
9%5E2%2Fa%5E2 %22-%2216%5E2%2F4a%5E2=1
81%2Fa%5E2 %22-%22256%2F4a%5E2=1
Multiplying both sides of the equal sign times 4a%5E2 ,
we get 4%2A81-256=4a%5E2
324-256=4a%5E2
68=4a%5E2
a%5E2=68%2F4=17
So, the equation we were looking for is
highlight%28x%5E2%2F17-y%5E2%2F68=1%29 .