JEE-Advanced 2015 (24/40)

Geometry Level 3

Suppose that the foci of the ellipse \(\frac{x^2}{9}+\frac{y^2}{5}=1\) are \((f_1,0)\) and \((f_2,0)\) where \(f_1>0\) and \(f_2<0\). Let \(P_1\) and \(P_2\) be two parabolas with a common vertex at \((0,0)\) and with foci at \((f_1,0)\) and \((2f_2,0)\) respectively. Let \(T_1\) be a tangent to \(P_1\) which passes through \((2f_2,0)\) and \(T_2\) be a tangent to \(P_2\) which passes through \((f_1,0)\). If \(m_1\) is the slope of \(T_1\) and \(m_2\) is the slope of \(T_2\), then find the value of \[\left( \frac{1}{m_1^2}+m_2^2 \right)\]

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