|
Question 114544: i need help with the following problem.
textbook:0-13-187027-0
chapter 7.2 question 82
"find the equation of a hyperbola whose asymptotes are perpendicular."
that's it. i've tried to figure it out but have been unable. any help would be greatly appreciated.
thanks
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Note: Remember, perpendicular lines can be found by multiplying the slopes. So if you had a slope of  and  , you just multiply to get  . Since the product is -1, the two lines are perpendicular.
Let's use the general equation of a hyperbola 
Remember, the asymptotes are simply equations of lines in the form of  . Also, their slopes  can be found by the ratio of a and b. In other words,  or  . Now multiply these two slopes to get
Since the two asymptotes are perpendicular, their product is equal to -1. In other words, 
 Multiply both sides by 
 Multiply both sides by  to get rid of the negatives
 Take the square root of both sides
So this means that when a is equal to b, the two slopes are perpendicular
Now let a=1, then b=1 (you can pick any value). Now simply plug this into  (note: I'm going to let h and k equal zero since they don't affect the solution)
which simplifies to
If we graphed this, we would get
and we can see that the asymptotes are the equations  and  which are perpendicular.
|
|
|
| |
