Definition of derivative of F(x) at a given point : It's the slope of the tangent to the graph of F(x) at that point.
So find the derivatives at t and -t, to obtain the slope of the respective tangents at the two points.
Now two lines are perpendicular if their slopes m1, m2 (say.) are such that (m1)(m2) = -1
Use this to find t in terms of k.
Note that you'll have to solve a quadratic equation in t.