Find the equation of a hyperbola which is generated by a point that moves so that the difference of its distance from the points (-4,1) and (2,1) is 4.

Answer 1

let (x,y) be any pt on the hyperbola
from your condition

√[(y-1)^2+(x+4)^2] - √[(y-1)^2+(x-2)^2] = 4
I moved the second radical to the right side, squared each side and simplified to get
3x-1 = √[(y-1)^2+(x-2)^2]
squaring again and simplifying I get

5x^2 + 10x - 4y^2 + 8y = 19

after completing the square I ended up with the standard form of

(x+1)^2/4 - (y-1)^2/5 = 1