parabola help!!

Question

When waiting to land, an airplane is traveling above the airport in a holding pattern that can be modeled by the equation 49x^2 + 64y^2 = 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)x^2. Should the air traffic controller at the airport be concerned about a collision
If yes, what are the possible points of collision?

Answer 1

49x^2 + 64y^2 = 3136

x^2/8^2 + y^2/7^2 = 1

is an ellipse with axes 16 and 14, with center at the origin

y-6 = (1/4)x^2 is a parabola with vertex = maximum point = (0,6)

it helps to draw a rough sketch

they intersect from a rough sketch about (+/- 2 or so, 6.5)

but to be more precise, try a graphing calculator

it shows about (-1.81,6.819) and (1.81,6.819) as the possible collusion points

or work out the algebra,

solve the ellipse and parabola equations for y, set them equal

solve for x

it depends on when the planes enter their orbit paths.

there may be no possibility of collision, but if there is, i

t's the above two points. also the smaller the planes

the less likely a collision.