Parth's Parabola

Calculus Level 5

Consider the section SS of the parabola y=2x2y = 2 - x^{2} lying in the first quadrant. What is the minimum possible length LL of a line segment (in the first quadrant) that is tangent to SS and has one endpoint lying on the xx-axis and the other on the yy-axis?

If L=m+nn128L = \sqrt{\dfrac{m + n\sqrt{n}}{128}} where m,nm,n are positive integers with nn square-free, then find m+n.m + n.

Comments: Happy Birthday, Parth Lohomi. :)

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