Question

The orbit of Pluto can be modeled by the equation {x^2}/{39.5^2} + {y^2}/{38.3^2 } = 1, where the units are astronomical units. Suppose a comet is following a path modeled by the equation x = y^{2} + 20. What are the points of intersection of the orbits of Pluto and the comet? (Not necessarily where they'll hit each other, just where the paths they take cross each other.)

Answer 1

9514 1404 393

Answer:

(39.24, -4.386), (39.24, 4.386)

Explanation:

A graphing calculator solves this easily.

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For an algebraic solution, you can substitute for y^2. To avoid messing with large numbers, we define p=39.5^2 and q=38.3^2. Then after substitution for y^2, we have ...

x^2/p +(x-20)/q = 1

Multiplying by pq gives ...

qx^2 +px -20p = pq

qx^2 +px -20p -pq = 0

x = (-p +√(p^2 +4qp(p+q)))/(2q)

Putting the numbers back into this equation gives ...

x ≈ 39.2401

y = ±√(x -20) = ±4.38635

The crossing points are (39.2401, ±4.38635).