Lines Tangent to a Circle

Geometry Level 4

There are two lines of slope \(- \frac{2}{5} \) that are tangent to a circle of radius \(1\) centered at the origin \((0,0)\). Only one of these lines has a positive \(y\)-intercept. Let the point of tangency of this line to the circle be \((x,y)\). When \(x+y\) is written in the form \(\frac{a \sqrt{b}}{c},\) where \(\gcd(a,c)=1\) and \(b\) is not divisible by the square of any prime, what is \(a+b+c?\)

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