JEE (Advanced) 2016 Paper-2

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Football teams $$T_1$$ and $$T_2$$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $$T_1$$ winning, drawing and losing a game against $$T_2$$ are $$\frac{1}{2}, \frac{1}{6}$$ and $$\frac{1}{3}$$ respectively. Each teamgets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let X and Y denote the total points scored by teams $$T_1$$ and $$T_2$$ respectively, after two games.

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Let $$F_1(x_1, 0)$$ and $$F_2(x_2, 0)$$, for $$x_1 < 0$$ and $$x_2 > 0$$, be the foci of the ellipse $$\frac{x^2}{9} + \frac{y^2}{8} = 1$$. Suppose a parabola having vertex at the origin and focus at $$F_2$$ intersects the ellipse at point M in the first quadrantandat point N in the fourth quadrant.