im having trouble with some math problems can you help

Question

the question is write the equation of the hyperbola with the given foci and vertices.  foci:(6,0), (-6,0)  vertices: (4,0)(-4,0)

Answer 1

Hello,

 

This is an East - West hyperbola centered at the origin.    The standard form is (x/a)² - (y/b)² = 1

 

With this standard form, the distance from the center to a vertex is a , so   a = 4.    The distance from the center to a focal point is:

sqrt( a² + b²)   ,  so    6 =  sqrt( 16 + b²)  .    Solving this for b  results in b =  sqrt(20).

 

Plugging these values for a and b into the standard form results in

(x/4)² - (y/sqrt(20))²  = 1

 

The only practical way to deal with hyperbola or ellipse problems like this one is to memorize the formulas for  standard forms and distances to vertex, foci and directrix.