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)
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.