Find the Equation of a Shifted Conic Find an equation for th

Find the Equation of a Shifted Conic. Find an equation for the conic section with the given properties.

The hyperbola with verticies V₁(-1,-1) and V₂(5,-1) and foci F₁(-4,-1) and F₂(8,-1)

Solution

vertices are (-1,-1)   and (5,-1)

And focii are (-4,-1) and (8,-1)

This hyperbola is of the form

(x-h)²/a²- (y-k)²/b²=1

Where vertices = (h+-a,k)

Therefore h+a=5,h-a=-1   and   k=-1

From these two equations we get

h=2 and a=3

And focii are (h+-c,k)

Therefore h+c=8 , h-c=-4

From these two equations we get

c= 6

And c²=a²+b²

      36= 9 + b²

b²=27

Therefore the required equation is

(x-2)²/ 9 - (y+1)²/27=1