Introduction:
To determine when the line y=2x will touch the parabola x^2=8y, we need to find the point(s) of intersection between the line and the parabola. The line can touch the parabola in one or two points, or it may not touch the parabola at all.

Step 1: Set up the equations:
The equation of the line is y=2x, and the equation of the parabola is x^2=8y. To find the point(s) of intersection, we can substitute y=2x into the equation of the parabola.

Step 2: Substitute y=2x into the parabola equation:
Substituting y=2x into x^2=8y gives:
x^2=8(2x)
x^2=16x

Step 3: Solve for x:
Rearranging the equation, we have:
x^2 - 16x = 0

Step 4: Factorize or use the quadratic formula:
Factoring the equation or using the quadratic formula, we find that x=0 or x=16.

Step 5: Find the corresponding y-values:
Substituting x=0 into the equation of the line y=2x, we find y=0. So one point of intersection is (0,0).
Substituting x=16 into the equation of the line y=2x, we find y=32. So the other point of intersection is (16,32).

Step 6: Conclusion:
Therefore, the line y=2x will touch the parabola x^2=8y at the points (0,0) and (16,32).

Summary:
In summary, to determine when the line y=2x will touch the parabola x^2=8y, we substitute y=2x into the equation of the parabola. Solving for x, we find x=0 and x=16. Substituting these values back into the equation of the line, we find the corresponding y-values. Therefore, the line touches the parabola at the points (0,0) and (16,32).

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