Given 25x^2 - 50x - 4y^2 + 24y = 111
we start moving to a standard form as follows
25(x^2 - 2x) - 4(y^2 + 6y) = 111
completing the square for both x and y portions gives us
25(x^2 - 2x + 1) - 25 - 4(y^2 + 6y + 9) + 36 = 111
this simplifies to
25(x - 1)^2 - 4(y + 3)^2 = 111 + 25 - 36
25(x - 1)^2 - 4(y + 3)^2 = 100
Divide both sides by 100 to get
(x - 1)^2 / 4 - (y + 3)^2 / 25 = 1
Since this is in the form (x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
the conic section is a hyperbola where h = 1, k = -3, a = 2, b = 5
done!