SOLUTION: Write an equation for the hyperbola that satisfies each set of conditions. vertices (9,-3) and (-5,-3), foci at (2(+or-)the square root of 53,-3)

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Write an equation for the hyperbola that satisfies each set of conditions. vertices (9,-3) and (-5,-3), foci at (2(+or-)the square root of 53,-3)      Log On


   

Question 42147This question is from textbook Algebra 2
: Write an equation for the hyperbola that satisfies each set of conditions.
vertices (9,-3) and (-5,-3), foci at (2(+or-)the square root of 53,-3) This question is from textbook Algebra 2

Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!

Write an equation for the hyperbola that satisfies
each set of conditions.                  __                   
vertices (9,-3) and (-5,-3), foci at (2ｱﾖ53,-3)

This is a hyperbola that looks like this:   }{

and has equation

(x-h)ｲ   (y-k)ｲ 
覧覧覧 - 覧覧覧 = 1
  aｲ       bｲ 

where center = (h,k),
a = semi-transverse axis, 
b = semi conjugate axis,
foci (hｱc,k) where cｲ = aｲ+bｲ,
and asaymptotes y - k = ｱ(b/a)(x - h)

The transverse axis is the line that goes from one vertex to the other.
The distance between (9,-3) and (-5,-3) is 14 units.
Therefore the semi-transverse axis, a = 7 units.

So far we have:

(x-h)ｲ   (y-k)ｲ 
覧覧覧 - 覧覧覧 = 1
  7ｲ       bｲ 

The center is midway between the foci, and since the midpoint between
(9,-3) and (-5,-3) is (2,-3), that is the center (h,k)
h = 2, k = -3

So far we have:

(x-2)ｲ   (y+3)ｲ 
覧覧覧 - 覧覧覧 = 1
  7ｲ       bｲ 

We only need b.
                          __
and we are given that (2ｱﾖ53,-3) are the foci.
We know that the foci = (hｱc,k), or (2ｱc,-3), so
c = ﾖ53

Also we know that
                               __
cｲ = aｲ+bｲ, and a = 7 and c = ﾖ53
  __
(ﾖ53)ｲ = 7ｲ + bｲ

  53 = 49 + bｲ
   4 = bｲ
   b = 2

So now we have b and the hyperbola is

(x-2)ｲ   (y+3)ｲ 
覧覧覧 - 覧覧覧 = 1
  7ｲ       2ｲ

or


(x-2)ｲ   (y+3)ｲ 
覧覧覧 - 覧覧覧 = 1
  49       4


      
     hyperbola only                   with asymptotes 

Edwin
AnlytcPhil@aol.com