3.5b Nature of the Roots (Combination of Straight Line and a Curve)

Example 2 (Combination of Straight Line and a Curve)
The straight line $y=2k+1$   intersects the curve $y=x+\frac{k^2}{x}$   at two distinct points.  Find the range of values of k.

Example 3 (Straight Line does not intersect the curve)
Find the range of values of m for which the straight line $y=mx+6$   does not meet the curve $2x^2-xy=3$  .