# Shortest Length of Line Intersection

If P is any point on the parabola y = x^2 except for the origin, let Q be the point where the normal line intersects the parabola again. Find the shortest possible length of the line segment PQ.

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If P is any point on the parabola y = x^2 except for the origin, let Q be the point where the normal line intersects the parabola again. Find the shortest possible length of the line segment PQ.

Solution. Suppose that is a point on the parabola except for the origin. So we have

Now we know that the tangent line L at P(a,b) can be determined as follows. First, the slope of L is

So, we can get the slope of the normal line L' at P(a,b) as follows.

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#### Solution Summary

This shows how to find the shortest possible length of a segment from a parabola to an intersection with the normal line.