Find the points on the hyperbola `x ^ 2-y ^ 2 = 8`  whose distance from the center of the hyperbola equally linear eccentricity.

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tiburtius | High School Teacher | (Level 2) Educator

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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).`