SOLUTION: A hyperbola has a vertical transverse axis of length 14 and asymptotes of y=9/8x + 1 and y= -9/8x - 9. Find the center of the hyperbola, its focal length, and its eccentricity.

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: A hyperbola has a vertical transverse axis of length 14 and asymptotes of y=9/8x + 1 and y= -9/8x - 9. Find the center of the hyperbola, its focal length, and its eccentricity.       Log On


   

Question 1039199: A hyperbola has a vertical transverse axis of length 14 and asymptotes of
y=9/8x + 1 and y= -9/8x - 9. Find the center of the hyperbola, its focal length, and its eccentricity.
Can you please show me the work to get the center mainly?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The center is the intersection of the asymptotes.
That is the point that is a solution to system%28y=%289%2F8%29x%2B1%2Cy=-%289%2F8%29x-9%29 .
You can solve that system of equations by substitution.
For example, you could start by substituting the expression %289%2F8%29x%2B1 for y in y=-%289%2F8%29x-9 .
system%28y=%289%2F8%29x%2B1%2Cy=-%289%2F8%29x-9%29 --> system%28y=%289%2F8%29x%2B1%2C%289%2F8%29x%2B1=-%289%2F8%29x-9%29 --> system%28y=%289%2F8%29x%2B1%2C%289%2F8%29x%2B%289%2F8%29x=-9-1%29 --> system%28y=%289%2F8%29x%2B1%2C%289%2F4%29x=-10%29 --> system%28y=%289%2F8%29x%2B1%2Cx=-10%284%2F9%29%29 --> system%28y=%289%2F8%29x%2B1%2Cx=-40%2F9%29 ,
and then, you could substitute the value found for one variable into one of the equations, to find the value of the other variable:
system%28y=%289%2F8%29x%2B1%2Cx=-40%2F9%29 --> system%28y=%289%2F8%29%28-40%2F9%29%2B1%2Cx=-40%2F9%29 --> system%28y=%289%2F8%29%28-40%2F9%29%2B1%2Cx=-40%2F9%29 --> system%28y=-5%2B1%2Cx=-40%2F9%29 --> highlight%28system%28y=-4%2Cx=-40%2F9%29%29

The asymptotes slopes give you the b%2Fa=9%2F8 ratio of conjugate to transverse axes.
The transverse axis is the segment joining the vertices of the hyperbola.
If the length is 14 , the distance from the center to each vertex must be
a=14%2F2=7 .
From there you can find b , c=sqrt%28b%5E2%2Ba%5E2%29 , and the eccentricity: c%2Fa .