Question

When waiting to land, an airplane is traveling above the airport in a holding pattern that can be modeled by the equation 49x2 + 64y2 = 3136, with the air traffic control tower at the origin. Suppose that another plane approaches the airport at the same altitude as the first plane on a path that can be modeled by the equation y - 6 = (1/4)x2. what are the possible points of collision?

Answer 1

49x^2 + 64y^2 = 3136 . . . (1)
y - 6 = (1/4)x^2 . . . (2)

From (2), x^2 = 4(y - 6) = 4y - 24 . . . (3)

Putting (3) into (1) gives
49(4y - 24) + 64y^2 = 3136
196y - 1176 + 64y^2 = 3136
64y^2 + 196y - 4312 = 0
y = 6.819
x = sqrt(4(6.819) - 24)
x = 1.81

The possible point of collision is (1.81, 6.819)