# Line Relection

Calculus Level 5

Let line $${\bf L}$$ be the tangent line to the curves $${\bf f(x) = -x^2 }$$ and $${\bf g(x) = \sqrt{x}. }$$ 

If $${\bf f(x) = -x^2 \: , \: }$$ $${\bf g(x) = \sqrt{x} }$$ and line $${\bf L}$$ are reflected about the line $${\bf y = \frac{1}{\sqrt{3}} x \:, }$$ then the image of $${\bf f(x) = -x^2 \: , \: }$$ $${\bf g(x) = \sqrt{x} }$$ and line $${\bf L}$$ can be expressed as $${\bf x^2 + a\sqrt{b} xy + b y^2 + a\sqrt{b} x - a y = 0 \: , }$$  $${\bf b x^2 - a\sqrt{b} xy + y^2 - a x - a\sqrt{b} y = 0 \: ,}$$  and $${\bf a (1 - \sqrt{b}) x + a (1 + \sqrt{b}) y + 1 = 0 }$$ respectively, where $${\bf a }$$ and $${\bf b }$$ are coprime positive integers.



Find: $${\bf a + b. }$$

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