Question 851093
A point moves so that the absolute value of the difference between it's distances from (0,5) and (0,-5) is 8.
Conic described is a hyperbola with a horizontal transverse axis, foci at (0,5) and (0,-5), 2a=8, and center at (0,0).
Its standard form of equation: {{{x^2/a^2-y^2/b^2=1}}}
2a=8
a=4
a^2=16
c=5
c^2=25
c^2=a^2+b^2
b^2=c^2-a^2=25-16=9
equation:  {{{x^2/16-y^2/9=1}}} 
see graph below as a visual check:
y=(9x^2/16-9)^.5

{{{ graph( 300, 300, -10, 10, -10, 10,(9x^2/16-9)^.5,-(9x^2/16-9)^.5) }}}