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anonymous

  • 5 years ago

Prove that the area bounded by a tangent at P(cp, c/p) to the rectangular hyperbola xy = c² and the asymptotes of the curve is a constant (ie, not dependent on p) Help please?

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  1. anonymous
    • 5 years ago
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    Asymptotes are the x and y axis, and the tangent at P is x + p²y = 8p Other than that, I don't know how to do it

  2. myininaya
    • 5 years ago
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    this sounds like you may need integration? do you know integration?

  3. anonymous
    • 5 years ago
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    Yep. I just don't know how to apply it to this question

  4. anonymous
    • 5 years ago
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    as x approaches infinity, the hyperbola approaches 0, but it doesn't touch, so I'm not sure what I should integrate since I only have the lower limit

  5. myininaya
    • 5 years ago
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    well that tangent line touches the curve sometime after x=0 it touches the tangent at x=cp so assume cp>0 and use as upper limit integrate tanget line - y=c^2/x from 0 to cp and see what happens

  6. anonymous
    • 5 years ago
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    oh! wait hold on a sec lemme draw something

  7. anonymous
    • 5 years ago
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  8. anonymous
    • 5 years ago
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    am I trying to find the yellow bit? I was trying to work out the area of the green bit and... yeah, it didn't work

  9. myininaya
    • 5 years ago
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    oh thanks for visual i like visuals

  10. anonymous
    • 5 years ago
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    lol :)

  11. myininaya
    • 5 years ago
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    you need to find the x-intercept of the tangent line

  12. anonymous
    • 5 years ago
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    alright, I think I know how to do this now. Thanks!

  13. myininaya
    • 5 years ago
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    i will do it too and we can compare answers k :)

  14. anonymous
    • 5 years ago
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    okay :) i'll come back when I'm done

  15. anonymous
    • 5 years ago
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    oh... i'm using x + p²y = 8p as the equation of the tangent... that's not right is it... I just realised that I got that from a question where c² = 16 so do I have to work out a general one?

  16. myininaya
    • 5 years ago
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    i didnt check this but you said to find tangent at (cp,c/p) right ?

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  17. myininaya
    • 5 years ago
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    ok i think this is right lol

  18. myininaya
    • 5 years ago
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    are you looking at what i did rain?

  19. anonymous
    • 5 years ago
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    yep, trying to figure it out

  20. anonymous
    • 5 years ago
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    I integrated the wrong thing. I put the equation of the hyperbola instead

  21. myininaya
    • 5 years ago
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    right according to your pic we are just wanting the area under that tangent line

  22. myininaya
    • 5 years ago
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    from x=0 to x=2cp

  23. anonymous
    • 5 years ago
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    y=c^2 /x =c^2 x^(-1) slope m= tangent line=-c^2 x^(-2)= -c^2 /x^2 the slope at P(cp, c/p) m=-c^2 / c^2p^2=1/p^2

  24. anonymous
    • 5 years ago
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    ah, okay, I got the answer :) Awesome, thanks so much!!

  25. myininaya
    • 5 years ago
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    you got 2c^2?

  26. anonymous
    • 5 years ago
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    I haven't finished yet, on the third last line. it's just algebra, so it should be right

  27. anonymous
    • 5 years ago
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    okay, got the answer :D Thank you!!!

  28. myininaya
    • 5 years ago
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    on that attachemnt i started my work in the second colum n and finished upp inthe first

  29. myininaya
    • 5 years ago
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    ok cool rain :)

  30. anonymous
    • 5 years ago
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    since m=slope=-c^2/x^2 you can get the EQ of the line y-y1=m(x-x1) y- c/p = (-c^2/ x^2)(x-cp) y = -c^2/x +c^3 p/x^2 +c/p you can now do the area by integration here

  31. anonymous
    • 5 years ago
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    good luck Rain, hope you get the correct answer

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