Suppose that the foci of the ellipse x^2 / 9 + y^2 / 5 = 1 are (f1, 0) and (f2, 0) where f1 > 0 and f2 < 0.

Question

Suppose that the foci of the ellipse \(\frac{x^2}{9}+\frac{y^2}{5}\) = 1 are (f₁, 0) and (f₂, 0) where f₁ > 0 and f₂ < 0. Let P₁ and P₂ be two parabolas with a common vertex at (0, 0) and with foci at (f₁, 0) and (2f₂, 0), respectively. Let T₁ be a tangent to P₁ which passes through (2f₂, 0) and T₂ be a tangent to P₂ which passes through (f₁, 0). The m₁ is the slope of T₁ and m₂ is the slope of T₂, then the value of \((\frac{1}{m^2}+m^2_2)\)

Answer 1

The equation of P₁ is y² – 8x = 0 and P₂ is y² + 16x = 0 

Tangent to y² – 8x = 0 passes through (–4, 0)

Also tangent to y² + 16x = 0 passes through (2, 0)