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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