Find the points on the hyperbola `x ^ 2-y ^ 2 = 8` whose distance from the center of the hyperbola equally linear eccentricity.
General equation of hyperbola is
where point `C(p,q)` is center of hyperbola.
If we divide equation of our hyperbola we get
From this we see that center is `C(0,0)` and `a^2=b^2=8.`
Now we need to calculate linear eccentricity for which we can use the following formula
Hence we have
Now all points whose distance from center `C` is `e` lie on a circle with center in `C(0,0)` and with radius `e.` Since general equation of circle is
equation of our circle is
Now we only need to find points that lie on both circle and hyperbola and for that we need to solve the following system of equations
If we add those two equations we get
Now we put that into second equation
So your solutions are `(-2sqrt3,-2),` `(-2sqrt3,2),` `(2sqrt3,-2)` and `(2sqrt3,2).`