Suppose that the foci of the ellipse x²/9 + y²/5 = 1 are (f1, 0) and (f2, 0) where f1 > 0 and f2 < 0. Let P1 and P2 be the 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). If m1 is the slope of T1 and m2 is the slope of T2, then the value of (1/m1² + m2²) is

Given ellipse is x²/9 + y²/5 = 1
e² = 1 - b²/a² = 1 - 4/9 = 4/9 => e = 2/3
Foci of the ellipse are (±ae, 0)
So, (f1, 0) is (2, 0) and (f2, 0) is (-2, 0)
Foci of parabola P1 is (2, 0) and foci of P2 is (-4, 0).
Equation of tangent to any parabola in slope form is y = mx + a/m.
Therefore, the equation of tangent to P1 is y = m1x + 2/m1
It passes through (-4, 0). => 0 = -4m1 + 2/m1 => m1² = 1/2 => 1/m1² = 2.. (i)
Equation of tangent to P2 is y = m2x + a/m => y = m2x + (-4)/m2
=> 0 = 2m2 + (-4)/m2 => m2² = 2. (ii)
Adding (i) and (ii), we get
1/m1² + m2² = 2 + 2
1/m1² + m2² = 4
So, correct answer is 4.