Question

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the
hyperbola 49y2 - 16x2 = 784​

Answer 1

Answer:

Given :

Equation of hyperbola :

49 y² - 16 x² = 784

Dividing by 784 :

49 / 784 y² - 16 / 784 x² = 784 / 784

y² / 16  - x² / 49 = 1

y² / 4² - x² / 7² = 1

We have standard equation of hyperbola :

y² / a² - x² / b² = 1

On comparing we get :

a = 4 and b = 7

We know :

c = √ ( a² + b² )

c = √ ( 16 + 49 )

c = √ 65.

Now : coordinate of foci ( 0 , ± c )

=  > c ( 0,  ± √ 65 )

Vertices ( 0 , ± a  )

= > ( 0 ,  ± 4  )

Eccentricity e = c / a

e = √ 65 / 4

Length of lotus rectum = 2 b² / a

= > 2 × 7² / 4

= >  49 / 2  units .

Therefore we get all required answer.