# Parth's Parabola

Calculus Level 5

Consider the section $$S$$ of the parabola $$y = 2 - x^{2}$$ lying in the first quadrant. What is the minimum possible length $$L$$ of a line segment (in the first quadrant) that is tangent to $$S$$ and has one endpoint lying on the $$x$$-axis and the other on the $$y$$-axis?

If $$L = \sqrt{\dfrac{m + n\sqrt{n}}{128}}$$ where $$m,n$$ are positive integers with $$n$$ square-free, then find $$m + n.$$

Comments: Happy Birthday, Parth Lohomi. :)

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