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calyne

The equation x^2 - xy + y^2 = 3 represents a "rotated ellipse," that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses the x-axis and show that the tangent lines at these points are parallel.

  • 2 years ago
  • 2 years ago

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  1. dpaInc
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    to find the x-intercepts plug in y=0 and solve for x. to show that the tangent lines are parallel, plug in the coordinates of these points to the derivative y'. if they're parallel, they should have the same value...

    • 2 years ago
  2. calyne
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    wait so the x-intercepts are [+-]sqrt(3)?

    • 2 years ago
  3. dpaInc
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    yes

    • 2 years ago
  4. calyne
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    well i got y' = -2x/(-1+2y)

    • 2 years ago
  5. calyne
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    so what, -2*sqrt(3)/(-1+0) = 2*sqrt(3), and then the same for -sqrt(3)..? what's that then? -2*-sqrt(3)/(-1+0) = -2sqrt3 i don't get it

    • 2 years ago
  6. dpaInc
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    check your derivative... again... i got something different

    • 2 years ago
  7. calyne
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    oh right crap

    • 2 years ago
  8. calyne
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    ok y' = (-2x+y)/(-x+2y) ?

    • 2 years ago
  9. dpaInc
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    that's what i got..

    • 2 years ago
  10. calyne
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    so substituting (sqrt(3),0) m=2?

    • 2 years ago
  11. dpaInc
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    yep. notice you get the same thing for the other point... so they're parallel.

    • 2 years ago
  12. calyne
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    yup. gotcha. thanks

    • 2 years ago
  13. dpaInc
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    yw

    • 2 years ago
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