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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.)

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sqdancefan

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Answer:

  (39.24, -4.386), (39.24, 4.386)

Step-by-step 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).

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