SOLUTION: Find the equation of a hyperbola in the form y²/M - x²/N = 1, M, N> 0 if the center is at the origin, the length of the conjugate axis is 10, and the foci are sqrt(29) units from

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Find the equation of a hyperbola in the form y²/M - x²/N = 1, M, N> 0 if the center is at the origin, the length of the conjugate axis is 10, and the foci are sqrt(29) units from      Log On


   

Question 1183543: Find the equation of a hyperbola in the form y²/M - x²/N = 1, M, N> 0 if the center is at the origin, the length of the conjugate axis is 10, and the foci are sqrt(29) units from the center.
Found 3 solutions by MathLover1, Edwin McCravy, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

y%5E2%2FM+-+x%5E2%2FN+=+1, M, N%3E+0 if the center is at the origin
if given the standard form y%5E2%2Fa%5E2-x%5E2%2Fb%5E2=1, then foci have the form (0,±c), and the transverse axis is the y-axis
=> c=sqrt%2829%29
the foci are at: (0 , sqrt%2829%29) and (0,-sqrt%2829%29 )
in your case
M=a%5E2
N=b%5E2
the length of the conjugate axis is 2a=10=>a=5=>a%5E2=25

c%5E2=a%5E2%2Bb%5E2
b%5E2=c%5E2-a%5E2
b%5E2=29-25
b%5E2=4

then
M=a%5E2=25
N=b%5E2=4
and your equation is:
y%5E2%2F25+-+x%5E2%2F4+=+1




Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
 I think she confused the conjugate axis and the transverse axis. It's
easy to do.

y%5E2%2Fa%5E2-x%5E2%2Fb%5E2=1 is a hyperbola that opens right and left.

b is the length of the semi-conjugate axis, so b = half of 10, so b = 5. 

y%5E2%2Fa%5E2-x%5E2%2F5%5E2=1
y%5E2%2Fa%5E2-x%5E2%2F25=1

a is the semi-transverse axis, which is the distance from the center to the vertex,
the foci are sqrt(29) units from the center.
c = the distance from the center to either of the foci.

The Pythagorean relation for all hyperbolas is c%5E2=a%5E2%2Bb%5E2

%28sqrt%2829%29%29%5E2=a%5E2%2B5%5E2

29=a%5E2%2B25

4=a%5E2

2=a

The equation is

y%5E2%2F4-x%5E2%2F25=1

 

Edwin

Answer by ikleyn(52797) About Me  (Show Source):
You can put this solution on YOUR website!
.

The solution by Edwin is correct.

The solution by @MathLover1 is not correct.


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Notice


    Edwin, please look into your first formula and the comment after it.


    I think this comment must be re-edited in this way


        " y%5E2%2Fa%5E2-x%5E2%2Fb%5E2=1 is a hyperbola that opens up and down. "


    If you agree, I will erase this notice as soon as I see your correction implemented in your post.